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Truth Tables, Tautologies, and Logical Equivalence

Mathematics normally works with a two-valued logic: Every statement is eitherTrue or False. You can use truth tables to determine the truth or falsity of a complicated statement based on the truth or falsity of its simple components.
A statement in sentential logic is built from simple statements using the logical connectives $\lnot$ , $\land$ , $\lor$ , $\ifthen$ , and $\iff$ . I'll construct tables which show how the truth or falsity of a statement built with these connective depends on the truth or falsity of its components.
Here's the table for negation:

$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & \cr & P & & $\lnot P$ & \cr height2pt & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & \cr & T & & F & \cr height2pt & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & \cr & F & & T & \cr height2pt & \omit & & \omit & \cr \noalign{\hrule} }}$

This table is easy to understand. If P is true, its negation $\lnot P$ is false. If P is false, then $\lnot P$ is {\it true}.
$P \land Q$ should be true when both P and Q aretrue, and false otherwise:

$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & P & & Q & & $P \land Q$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & F & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & T & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & F & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}$

$P \lor Q$ is true if either P is true or Q is true (or both). It's only false if both P and Q arefalse.

$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & P & & Q & & $P \lor Q$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & F & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & T & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & F & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}$

Here's the table for logical implication:

$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & P & & Q & & $P \ifthen Q$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & F & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & T & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & F & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}$

To understand why this table is the way it is, consider the following example:
"If you get an A, then I'll give you a dollar."
The statement will be true if I keep my promise and false if I don't.
Suppose it's true that you get an A and it'strue that I give you a dollar. Since I kept my promise, the implication is {\it true}. This corresponds to the first line in the table.
Suppose it's true that you get an A but it'sfalse that I give you a dollar. Since I didn'tkeep my promise, the implication is false. This corresponds to the second line in the table.
What if it's false that you get an A? Whether or not I give you a dollar, I haven't broken my promise. Thus, the implication can't be false, so (since this is a two-valued logic) it must be true. This explains the last two lines of the table.
$P \iff Q$ means that P and Q are equivalent. So the double implication is true if P and Q are both true or if P and Q are both false; otherwise, the double implication is false.

$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & P & & Q & & $P \iff Q$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & F & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & T & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & F & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}$

You should remember --- or be able to construct --- the truth tables for the logical connectives. You'll use these tables to construct tables for more complicated sentences. It's easier to demonstrate what to do than to describe it in words, so you'll see the procedure worked out in the examples.
Remarks. 1. When you're constructing a truth table, you have to consider all possible assignments of True (T) and False (F) to the component statements. For example, suppose the component statements are P, Q, and R. Each of these statements can be either true or false, so there are

possibilities.
When you're listing the possibilities, you should assign truth values to the component statements in a systematic way to avoid duplication or omission. The easiest approach is to use lexicographic ordering. Thus, for a compound statement with three components P, Q, and R, I would list the possibilities this way:

$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & P & & Q & & R & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & T & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & F & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & T & & F & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & T & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & T & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & F & & T & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & F & & F & & F & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}$

2. There are different ways of setting up truth tables. You can, for instance, write the truth values "under" the logical connectives of the compound statement, gradually building up to the column for the "primary" connective.
I'll write things out the long way, by constructing columns for each "piece" of the compound statement and gradually building up to the compound statement.

Example. Construct a truth table for the formula $\lnot P \land (P \ifthen Q)$ .

$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & P & & Q & & $\lnot P$ & & $P \ifthen Q$ & & $\lnot P \land (P \ifthen Q)$ & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & T & & T & & F & & T & & F & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & T & & F & & F & & F & & F & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & F & & T & & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & F & & F & & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}\quad\halmos$

A tautology is a formula which is "always true" --- that is, it is true for every assignment of truth values to its simple components. You can think of a tautology as a rule of logic.
The opposite of a tautology is acontradiction, a formula which is "always false". In other words, a contradiction is false for every assignment of truth values to its simple components.

Example. Show that $(P \ifthen Q) \lor (Q \ifthen P)$ is a tautology.
I construct the truth table for $(P \ifthen Q) \lor (Q \ifthen P)$ and show that the formula is always true.

$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & P & & Q & & $P \ifthen Q$ & & $Q \ifthen P$ & & $(P \ifthen Q) \lor (Q \ifthen P)$ & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & T & & T & & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & T & & F & & F & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & F & & T & & T & & F & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & F & & F & & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}$

The last column contains only T's. Therefore, the formula is a tautology.

Example. Construct a truth table for $(P \ifthen Q) \land (Q \ifthen R)$ .

$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr & P & & Q & & R & & $P \ifthen Q$ & & $Q \ifthen R$ & & $(P \ifthen Q) \land (Q \ifthen R)$ & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr & T & & T & & T & & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr & T & & T & & F & & T & & F & & F & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr & T & & F & & T & & F & & T & & F & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr & T & & F & & F & & F & & T & & F & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr & F & & T & & T & & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr & F & & T & & F & & T & & F & & F & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr & F & & F & & T & & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr & F & & F & & F & & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}\quad\halmos$

Example. Suppose
"

" is true.
" $\displaystyle \int f(x)\,dx = g(x) + C$ " is false.
"Calvin Butterball has purple socks" is true.
Determine the truth value of the statement

$(x > y \ifthen \int f(x)\,dx = g(x) + C) \ifthen\,\lnot (\hbox{Calvin Butterball has purple socks})$

For simplicity, let
P = "

".
Q = " $\displaystyle \int f(x)\,dx = g(x) + C$ ".
R = "Calvin Butterball has purple socks".
I want to determine the truth value of $(P \ifthen Q) \ifthen\,\lnot R$ . Since I was given specific truth values for P, Q, and R, I set up a truth table with a single row using the given values for P, Q, and R:

Therefore, the statement is true.

Two statements X and Y are logically equivalent if $X \iff Y$ is a tautology. Another way to say this is: For each assignment of truth values to the simple statements which make up X and Y, the statements X and Y have identical truth values.
From a practical point of view, you can replace a statement in a proof by any logically equivalent statement.
To test whether X and Y are logically equivalent, you could set up a truth table to test whether $X \iff Y$ is a tautology --- that is, whether $X \iff Y$ "has all T's in its column". However, it's easier to set up a table containing X and Y and then check whether the columns for X and for Y are the same.

Example. Show that $P \ifthen Q$ and $\lnot P \lor Q$ are logically equivalent.

$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & P & & Q & & $P \ifthen Q$ & & $\lnot P$ & & $\lnot P \lor Q$ & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & T & & T & & T & & F & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & T & & F & & F & & F & & F & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & F & & T & & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr & F & & F & & T & & T & & T & \cr height2pt & \omit & & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}$

Since the columns for $P \ifthen Q$ and $\lnot P \lor Q$ are identical, the two statements are logically equivalent. This tautology is calledConditional Disjunction. You can use this equivalence to replace a conditional by a disjunction.

There are an infinite number of tautologies and logical equivalences; I've listed a few below; a more extensive list is given at the end of this section.

$\matrix{ \hbox{Double negation} & \lnot(\lnot P) \iff P \cr \hbox{DeMorgan's Law} & \lnot(P \lor Q) \iff (\lnot P \land \lnot Q) \cr \hbox{DeMorgan's Law} & \lnot(P \land Q) \iff (\lnot P \lor \lnot Q) \cr \hbox{Contrapositive} & (P \ifthen Q) \iff (\lnot Q \ifthen\,\lnot P) \cr \hbox{Modus ponens} & [P \land (P \ifthen Q)] \ifthen Q \cr \hbox{Modus tollens} & [\lnot Q \land (P \ifthen Q)] \ifthen\,\lnot P \cr}$

When a tautology has the form of a biconditional, the two statements which make up the biconditional are logically equivalent. Hence, you can replace one side with the other without changing the logical meaning.

Example. Write down the negation of the following statements, simplifying so that only simple statements are negated.
(a) $(P \lor \lnot Q)$

$\matrix{\lnot(P \lor \lnot Q) & \iff & \lnot P \land \lnot\lnot Q & \hbox{DeMorgan's law} \cr & \iff & \lnot P \land Q & \hbox{Double negation} \cr}\quad\halmos$

(b) $(P \land Q) \ifthen R$

$\matrix{\lnot[(P \land Q) \ifthen R] & \iff & \lnot[\lnot(P \land Q) \lor R] & \hbox{Conditional Disjunction} \cr & \iff & \lnot\lnot(P \land Q) \land \lnot R & \hbox{DeMorgan's law} \cr & \iff & (P \land Q) \land \lnot R & \hbox{Double negation} \cr}$

I showed that $(A \ifthen B)$ and $(\lnot A \lor B)$ are logically equivalent in an earlier example.

Example. Use DeMorgan's Law to write thenegation of the following statement, simplifying so that only simple statements are negated:
"Calvin is not home or Bonzo is at the movies."
Let C be the statement "Calvin is home" and let B be the statement "Bonzo is at the moves". The given statement is $\lnot C \lor B$ . I'm supposed to negate the statement, then simplify:

$\matrix{\lnot(\lnot C \lor B) & \iff & \lnot\lnot C \land \lnot B & \hbox{DeMorgan's Law} \cr & \iff & C \land \lnot B & \hbox{Double negation} \cr}$

The result is "Calvin is home and Bonzo is not at the movies".

Example. Use DeMorgan's Law to write thenegation of the following statement, simplifying so that only simple statements are negated:
"If Phoebe buys a pizza, then Calvin buys popcorn."
Let P be the statement "Phoebe buys a pizza" and let C be the statement "Calvin buys popcorn". The given statement is $P \ifthen C$ . To simplify the negation, I'll use theConditional Disjunction tautology which says

$(P \ifthen Q) \iff (\lnot P \lor Q)$

That is, I can replace $P \ifthen Q$ with $\lnot P \lor Q$ (or vice versa).
Here, then, is the negation and simplification:

$\matrix{\lnot(P \ifthen C) & \iff & \lnot(\lnot P \lor C) & \hbox{Conditional Disjunction} \cr & \iff & \lnot\lnot P \land \lnot C & \hbox{DeMorgan's Law} \cr & \iff & P \land \lnot C & \hbox{Double negation} \cr}$

The result is "Phoebe buys the pizza and Calvin doesn't buy popcorn".

Example. Replace the following statement with its contrapositive:
"If x and y are rational, then

is rational."
By the contrapositive equivalence, this statement is the same as "If

is not rational, then it is not the case that both x and y are rational".

Example. Show that the inverse and the converse of a conditional are logically equivalent.
Let $P \ifthen Q$ be the conditional. The inverse is $\lnot P \ifthen\,\lnot Q$ . The converse is $Q \ifthen P$ .
I could show that the inverse and converse are equivalent by constructing a truth table for $(\lnot P \ifthen\,\lnot Q) \iff (Q \ifthen P)$ . I'll use some known tautologies instead.
Start with $\lnot P \ifthen\,\lnot Q$ :

$\matrix{\lnot P \ifthen\,\lnot Q & \iff & \lnot\lnot Q \ifthen \lnot\lnot P & \hbox{Contrapositive} \cr & \iff & Q \ifthen P & \hbox{Double negation} \cr}$

Remember that I can replace a statement with one that is logically equivalent. For example, in the last step I replaced $\lnot\lnot Q$ with Q, because the two statements are equivalent by Double negation.

Example. Suppose x is a real number. Consider the statement
"If

, then

."
Construct the converse, the inverse, and the contrapositive. Determine the truth or falsity of the four statements --- the original statement, the converse, the inverse, and the contrapositive --- using your knowledge of algebra.
The converse is "If

, then

".
The inverse is "If $x^2 \ne 4$ , then $x \ne 2$ ".
The contrapositive is "If $x \ne 2$ , then $x^2 \ne 4$ ".
The original statement is false:

, but $-2 \ne 2$ . Since the original statement is eqiuivalent to the contrapositive, the contrapositive must be false as well.
The converse is true. The inverse is logically equivalent to the converse, so the inverse is true as well.

List of Tautologies

$\matrix{ 1. \hfill & P \lor \lnot P \hfill & \hbox{Law of the excluded middle} \hfill \cr 2. \hfill & \lnot(P \land \lnot P) \hfill & \hbox{Contradiction} \hfill \cr 3. \hfill & [(P \ifthen Q) \land \lnot Q] \ifthen \lnot P \hfill & \hbox{ Modus tollens} \hfill \cr 4. \hfill & \lnot\lnot P \iff P \hfill & \hbox{Double negation} \hfill \cr 5. \hfill & [(P \ifthen Q) \land (Q \ifthen R)] \ifthen (P \ifthen R) \hfill & \hbox{Law of the syllogism} \hfill \cr 6. \hfill & (P \land Q) \ifthen P \hfill & \hbox{Decomposing a conjunction} \hfill \cr & (P \land Q) \ifthen Q \hfill & \hbox{Decomposing a conjunction} \hfill \cr 7. \hfill & P \ifthen (P \lor Q) \hfill & \hbox{Constructing a disjunction} \hfill \cr & Q \ifthen (P \lor Q) \hfill & \hbox{Constructing a disjunction} \hfill \cr 8. \hfill & (P \iff Q) \iff [(P \ifthen Q) \land (Q \ifthen P)] \hfill & \hbox{Definition of the biconditional} \hfill \cr 9. \hfill & (P \land Q) \iff (Q \land P) \hfill & \hbox{Commutative law for $\land$} \hfill \cr 10. \hfill & (P \lor Q) \iff (Q \lor P) \hfill & \hbox{Commutative law for $\lor$} \hfill \cr 11. \hfill & [(P \land Q) \land R] \iff [P \land (Q \land R)] \hfill & \hbox{Associative law for $\land$} \hfill \cr 12. \hfill & [(P \lor Q) \lor R] \iff [P \lor (Q \lor R)] \hfill & \hbox{Associative law for $\lor$} \hfill \cr 13. \hfill & \lnot(P \lor Q) \iff (\lnot P \land \lnot Q) \hfill & \hbox{DeMorgan's law} \hfill \cr 14. \hfill & \lnot(P \land Q) \iff (\lnot P \lor \lnot Q) \hfill & \hbox{DeMorgan's law} \hfill \cr 15. \hfill & [P \land (Q \lor R)] \iff [(P \land Q) \lor (P \land R)] \hfill & \hbox{Distributivity} \hfill \cr 16. \hfill & [P \lor (Q \land R)] \iff [(P \lor Q) \land (P \lor R)] \hfill & \hbox{Distributivity} \hfill \cr 17. \hfill & (P \ifthen Q) \iff (\lnot Q \ifthen \lnot P) \hfill & \hbox{Contrapositive} \hfill \cr 18. \hfill & (P \ifthen Q) \iff (\lnot P \lor Q) \hfill & \hbox{Conditional disjunction} \hfill \cr 19. \hfill & [(P \lor Q) \land \lnot P] \ifthen Q \hfill & \hbox{Disjunctive syllogism} \hfill \cr 20. \hfill & (P \lor P) \iff P \hfill & \hbox{Simplification} \hfill \cr 21. \hfill & (P \land P) \iff P \hfill & \hbox{Simplification} \hfill \cr}$

Kids Worksheets

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English

English: Includes handwriting practice sheets, alphabet tracing, CVC words, vocabulary building activities, and more.

Handwriting practice sheets

Learn to Write

Fill In Missing Letters

Circle the Letters

Cursive Writing – Small Letters

Cursive Alphabet Trace

Trace 100 Sight Words

Write the First Letter of Given Picture

Circle the Correct Letter Worksheets

Circle the Cursive Letter Worksheets

Match the Letter with Correct Picture

Match the Picture with Cursive Letter

Matching Letters

Circle two pictures that begin with same letter sound

Circle two pictures that begin with same letter sound (Cursive)

CVC Worksheets Letter ‘a’

CVC Worksheets Letter ‘e’

CVC Worksheets Letter ‘i’

CVC Worksheets Letter ‘o’

CVC Worksheets Letter ‘u’

Look and write with vowels a, e, i, o, u

Opposites

Opposite Words

2 Letter words - sight words

Alphabet Minibooks

Literature.

Literature: Features nursery rhymes, limericks, stories, and reading passages.

Nursery Rhymes

Limericks

Cursive Alphabet Trace and Write

Letters A to G Upper and Lower Case Tracing Worksheet.

Cute Phrases A-Z

Sight Words

Beginning Sounds. Kindergarten Worksheet

Cursive Writing Small Letters.

Capital Letters.

Small Letters.

Alphabet Trace.

Alphabet Trace and Write.

Alphabet Worksheet

Consonant Vowel Consonant (CVC) Flashcards

Nursery Rhymes

Activities

Activities: Includes various activities related to numbers, princesses, Earth Day, animals, shapes, colors, and more.

Number Activities

Princess Activities

Earth Day Activities

Witches and Wizards Activities

Under the Sea Activities

Unicorn Activities

Transportation Activities

Reading Passages.

Reading Passages for Kids

Story PDF.

White Magic Story

Sunshine and Reeva in China

The Little Red Hen

The Sun,Moon and Wind

The Arab and the Camel

The Tortoise and the Hare

The Lion and the Mouse

Goldilocks and the Three Bears

The Three Little Pigs

The Princess and the Pea

The Shepherd Boy and the Wolf

Rapunzel

The Goose and the Golden Eggs

The North Wind and the Sun

The Miser and his Gold

The Country Maid and her Milk Pail

Goodnight Moon

The Ugly Duckling

The Boy Who Cried Wolf

Cinderella

Two Cats and Clever Monkey

The Lion and the Rabbit

The Lion and the Mouse

Mathematics.

Mathematics: Features number tracing, counting exercises, addition and subtraction worksheets, shape recognition activities, and other math-related resources.

Trace Numbers 1 to 10

Number Trace and Write

Adding Fun

Number Identification

Before,Between and After

Classroom Numbers

Measuring Things

Additional Worksheet.

Subtraction Worksheets

Same, Less, More

Counting

Count and Write

Count and Write Worksheets

Count and Match Worksheets

Count and Circle Worksheets

Fill in the Missing Number Worksheets

Dinosaur Math Activities

What Comes After & Between

Write Missing Numbers

Shape worksheets

Backward counting

Trace the numbers 1-10

Multiplication Sheet practice for Children

Counting practice from 1 to 100 Worksheet

Miscellaneous in Maths

Science.

Science: Offers science activity plans and worksheets.

Science

Science Activity Plans

Miscellaneous.

Animal Decorations

Classroom Decorations

Foldable Boxes

Teacher's Planner

Classroom Rules

Graduation Certificates

Placemats

UKG Worksheets

Geography.

Geography: Covers topics like weather and the calendar.

Geography

Weather

Calendar

Hindi

Hindi: Provides resources for learning Hindi alphabets (Swar and Vyanjan), tracing worksheets, vocabulary exercises on colors, fruits, vegetables, and more.

Hindi Alphabets. (Swar)

Hindi Alphabets. (Vanjan)

Colours name in Hindi | रंगों के नाम

Fruits name in Hindi | फलों के नाम

Vegetables name in Hindi | सब्जियों के नाम

Days in Hindi

Parts of Body

Hindi Swar Tracing Worksheets

Hindi Vyanjan Tracing Worksheets

Write the First Letter of picture - Hindi Swar Worksheets

Look and Match - Hindi Swar Worksheets

Circle the correct letter - Hindi Swar Worksheets

Write the first letter - Hindi Vyanjan Worksheets

Circle the Correct Letter - Vyanjan Worksheets

Choose the Right Image - Vyanjan Worksheets

Miscellaneous Hindi Worksheets

2 Letter Words Hindi Worksheets

3 Letter Words Hindi Worksheets

4 Letter Words Hindi Worksheets

AA (ा) – AA ki Matra | आ (ा) की मात्रा

i ( ि) - i ki Matra | इ ( ि) की मात्रा

EE ( ी) – EE ki Matra | ई ( ी) की मात्रा

U (ु) - U ki Matra | उ (ु) की मात्रा

O (ू ) – OO ki Matra | ऊ (ू) की मात्रा

E ( े) - E ki Matra | ए ( े ) की मात्रा

AI (ै) - AI ki Matra | ऐ (ै)की मात्रा

o ( ो) - o ki Matra | ओ (ो) की मात्रा

ou ( ौ) - ou ki Matra | औ ( ौ) की मात्रा

General Knowledge.

General Knowledge (GK): Includes worksheets on animal sounds, mazes, social skills, feelings, and other general knowledge topics.

GK Worksheets

Patterns What comes Next

Things That Go Together

Things That Do Not Belong

Match the following.

Match the fruit to its shadow. [5 Pages]

Match Letters [35 Pages]

Matching Worksheets

Sorting Worksheet

Shadow Matching

Match the uppercase letter to its lowercase [6 Pages]

Circle 2 Matching Pictures

Games.

Games: Offers puzzles, mazes, spot the difference games, Sudoku, and other fun activities.

Coloring.

Coloring: Provides coloring pages featuring animals, dinosaurs, alphabets, numbers, and more.

English Alphabet Color it.

English Alphabet Color it and Match it with Pictures.

Alphabet Color it. [26 Pages]

Alphabet Color it 2. [7 Pages]

English Alphabet Color it.

Numbers PDF.

Numbers 1 to 10 Color it. [2 Pages]

1 to 10 Numbers Coloring. [4 Pages]

Flash Cards PDF.

Flash Cards: Includes flash cards for plants, numbers, letters, shapes, colors, and other vocabulary words.

Plant Flashcards

Number Flashcards

Letters and Numbers

Tell the Time Flash Cards [6 Pages]

Flashcards English vocabulary [12 Pages]

Alphabet Letters with Pictures [5 Pages]

Numbers Flash Cards. [5 Pages]

Shapes FlashCards. [4 Pages]

Colors FlashCards. [3 Pages]

English Alphabet Learning Flash Cards. [26 Pages]

Alphabet Flashcards. [26 Pages]

Alphabet Identification Flash Cards. [26 Pages]

….

Addition

Addition Worksheet. [5 Pages] (V.1-5)

Addition Worksheet. [36 Pages] (V.1-5)

Additional Worksheet.

Subtraction

Subtracting by Pictures [5 Pages] (V.1-5)

Subtracting by Numbers [5 Pages] (V.1-5)

Subtracting by Pictures and Numbers [5 Pages] (V.1-5)

Subtract and circle the correct number [5 Pages] (V.1-5)

General Knowledge.

Fruits [6 Pages] (V.5)

Vegetables [6 Pages] (V.5)

Positions [7 Pages] (V.5)

Colors [10 Pages] (V.5)

Match the following.

Match the fruit to its shadow. [5 Pages] (V.1-5)

Match Letters [35 Pages] (V.1-5)

Match the uppercase letter to its lowercase [6 Pages] (V.1-5)

Mathematics.

Count and Write Worksheets

Count and Match Worksheets

Fill in the Missing Number Worksheets

Trace the numbers 1-10.

Multiplication Sheet practice for Children [14 Pages] (V.1-5)

Counting practice from 1 to 100 Kindergarten Math Worksheet

Games.

Freak - Out !!! [10 pages] (V.5)

Literature.

Nursery Rhymes

Cursive Alphabet Trace and Write [26 Pages] (V.1-5)

Letters A to G Upper and Lower Case Tracing Worksheet.

Beginning Sounds. Kindergarten Worksheet

Cursive Writing Small Letters. [7 Pages] (V.1-5)

Capital Letters. [26 Pages] (V.1-5)

Small Letters. [26 Pages] (V.1-5)
Alphabet Trace. [9 Pages] (V.1-5)

Alphabet Trace and Write. [26 Pages] (V.1-5)

Alphabet Worksheet [26 Pages] (V.1-5)

Consonant Vowel Consonant (CVC) Flashcards [33 Pages] (V.1-5)

Hindi PDF Download.

Hindi Alphabets. (Swar) [13 Pages] (V.1-5)

Hindi Alphabets. (Vanjan) [34 Pages] (V.1-5)

Story PDF Download.

Two Cats and Clever Monkey [5 pages] (V.1-5)

The Lion and the Rabbit [4 Pages] (V.1-5)

The Lion and the Mouse [2 Pages] (V.1-5)

Reading Passages PDF Download.

Reading Passages for Kids [5 Pages] (V.1-5)

Coloring PDF Download.

Alphabet Coloring. [26 Pages] (V.1-5)

Coloring Images. [12 Pages]

English Alphabet Color it. [5 Pages] (V.1-5)

English Alphabet Color it and Match it with Pictures. [5 Pages] (V.1-5)

Alphabet Color it. [26 Pages] (V.1-5)

Alphabet Color it 2. [7 Pages] (V.1-5)

English Alphabet Color it. 2 [5 Pages] (V.1-5)

Numbers PDF Download.

Numbers 1 to 10 Color it. [2 Pages] (V.1-5)

1 to 10 Numbers Coloring. [4 Pages] (V.1-5)

Flash Cards PDF Download.

Tell the Time Flash Cards [6 Pages] (V.5)

Flashcards English vocabulary [12 Pages] (V.5)

Alphabet Letters with Pictures [5 Pages] (V.5)

Numbers Flash Cards. [5 Pages] (V.1-5)

Shapes FlashCards. [4 Pages] (V.1-5)

Colors FlashCards. [3 Pages] (V.1-5)

English Alphabet Learning Flash Cards. [26 Pages] (V.1-5)

Alphabet Flashcards. [26 Pages] (V.1-5)

Alphabet Identification Flash Cards. [26 Pages] (V.1-5)

This document provides a ton of educational resources for kids. Here's a breakdown:

Subjects: English, Literature, Math, Science, Geography, Hindi, and General Knowledge.

English: Handwriting, alphabet tracing, CVC words, vocabulary.

Literature: Nursery rhymes, limericks, stories, reading passages.

Math: Number tracing, counting, addition, subtraction, shapes.

Science: Activity plans and worksheets.

Geography: Weather, calendar.

Hindi: Alphabet (Swar and Vyanjan), tracing, vocabulary (colors, fruits, vegetables).

General Knowledge: Animal sounds, mazes, social skills, feelings.

Other Fun Stuff: Coloring pages, puzzles, mazes, flash cards, and more!

Basically, it's a one-stop-shop for parents and teachers looking for engaging and educational materials for their kids.

This document is a treasure trove of worksheets and educational resources for kids!

It covers all sorts of subjects: English, Hindi, Math, Science, General Knowledge, and even Geography.

You'll find everything from handwriting practice to alphabet tracing, CVC words, vocabulary builders, and more.

There are also fun activities for numbers, princesses, Earth Day, animals, shapes, colors, and all sorts of other themes.

If you're looking for stories, there are nursery rhymes, limericks, and reading passages to enjoy.

Math whizzes will love the number tracing, counting exercises, addition and subtraction worksheets, and shape recognition activities.

Key Points - Worksheets for Kids

Subjects: English, Literature, Math, Science, Geography, Hindi, General Knowledge.

English:

Handwriting practice

Alphabet tracing

CVC words

Vocabulary building

Literature:

Nursery rhymes

Limericks

Stories

Reading passages.

Math:

Number tracing

Counting exercises

Addition and subtraction worksheets

Shape recognition.

Science:

Activity plans

Worksheets

Geography:

Weather

Calendar.

Hindi:

Alphabet practice (Swar and Vyanjan)

Tracing worksheets.

Vocabulary (colors, fruits, vegetables).

General Knowledge:

Animal sounds

Mazes

Social skills

Feelings.

Fun Activities:

Coloring pages

Puzzles

Mazes

Flashcards.

Remember: This is a super helpful resource for parents and teachers to find fun and educational materials for kids! Everything is organized and easy to print. Happy learning!

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The main topic of the document is providing a vast collection of worksheets and educational resources for kids. These resources cover various subjects like English, Hindi, Mathematics, Science, General Knowledge, and more. The document includes links to downloadable PDFs and webpages containing these materials.

The purpose of the "Worksheets for Kids New" document is to serve as a comprehensive directory or index of educational worksheets for children. It organizes links to a wide variety of worksheets, categorized by subject (English, Hindi, Mathematics, etc.) and type of activity (coloring, flash cards, stories, etc.). The document's purpose is to provide a convenient resource for parents, teachers, and caregivers to easily access and download these educational materials for kids.

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Pages

Truth Tables, Tautologies, and Logical Equivalence

List of Tautologies

Kids Worksheets