Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.4 [Pages 10 - 11]
Write the following statement in symbolic form.
If triangle is equilateral then it is equiangular.
Let p : Triangle is equilateral.
q : Triangle is equiangular.
The symbolic form is \( p \rightarrow q \).
Write the following statement in symbolic form.
It is not true that “i” is a real number.
Let p : i is a real number.
The symbolic form is \( \sim p \).
Write the following statement in symbolic form.
Even though it is not cloudy, it is still raining.
Let p : It is cloudy.
q : It is raining.
The symbolic form is \( \sim p \land q \).
Write the following statement in symbolic form.
Milk is white if and only if the sky is not blue.
Let p : Milk is white.
q : Sky is blue.
The symbolic form is \( p \leftrightarrow \sim q \).
Write the following statement in symbolic form.
Stock prices are high if and only if stocks are rising.
Let p : Stock prices are high.
q : Stock are rising.
The symbolic form is \( p \leftrightarrow q \).
Write the following statement in symbolic form.
If Kutub-Minar is in Delhi then Taj-Mahal is in Agra.
Let p : Kutub-Minar is in Delhi.
q : Taj-Mahal Is in Agra.
The symbolic form is \( p \rightarrow q \).
Find the truth value of the following statement.
It is not true that 3 − 7i is a real number.
Let p : 3 – 7i is a real number.
The truth value of p is F.
The given statement in symbolic form is \( \sim p \).
\( \therefore \sim p \equiv \sim F \equiv T \)
\( \therefore \) Truth value of the given statement is T.
Find the truth value of the following statement.
If a joint venture is a temporary partnership, then discount on purchase is credited to the supplier.
Let p : A joint venture is a temporary partnership.
q : Discount on purchase is credited to the supplier.
The truth value of p and q are T and F respectively.
The given statement in symbolic form is \( p \rightarrow q \).
\( \therefore p \rightarrow q \equiv T \rightarrow F \equiv F \)
\( \therefore \) Truth value of the given statement is F.
Find the truth value of the following statement.
Every accountant is free to apply his own accounting rules if and only if machinery is an asset.
Let p : Every accountant is free to apply his own accounting rules.
q : Machinery is an asset.
The truth values of p and q are F and T respectively.
The given statement in symbolic form is \( p \leftrightarrow q \).
\( \therefore p \leftrightarrow q \equiv F \leftrightarrow T \equiv F \)
\( \therefore \) Truth value of the given statement is F.
Find the truth value of the following statement.
Neither 27 is a prime number nor divisible by 4.
Let p : 27 is a prime number.
q : 27 is divisible by 4.
The truth values of p and q are F and F respectively.
The given statement in symbolic form is \( \sim p \land \sim q \).
\( \therefore \sim p \land \sim q \equiv \sim F \land \sim F \equiv T \land T \equiv T \)
\( \therefore \) Truth value of the given statement is T.
Find the truth value of the following statement.
3 is a prime number and an odd number.
Let p : 3 is a prime number.
q : 3 is an odd number.
The truth values of p and q are T and T respectively.
The given statement in symbolic form is \( p \land q \).
\( \therefore p \land q \equiv T \land T \equiv T \)
\( \therefore \) Truth value of the given statement is T.
If p and q are true and r and s are false, find the truth value of the following compound statement.
\( p \land (q \land r) \)
\( p \land (q \land r) \equiv T \land (T \land F) \)
\( \equiv T \land F \)
\( \equiv F \)
Hence, truth value is F.
If p and q are true and r and s are false, find the truth value of the following compound statement.
\( (p \rightarrow q) \lor (r \land s) \)
\( (p \rightarrow q) \lor (r \land s) \equiv (T \rightarrow T) \lor (F \land F) \)
\( \equiv T \lor F \)
\( \equiv T \)
Hence, truth value is T.
If p and q are true and r and s are false, find the truth value of the following compound statement.
\( \sim [(\sim p \lor s) \land (\sim q \land r)] \)
\( \sim [(\sim p \lor s) \land (\sim q \land r)] \equiv \sim[(\sim T \lor F) \land (\sim T \land F)] \)
\( \equiv \sim[(F \lor F) \land (F \land F)] \)
\( \equiv \sim (F \land F) \)
\( \equiv \sim F \)
\( \equiv T \)
Hence, truth value is T.
If p and q are true and r and s are false, find the truth value of the following compound statement.
\( (p \rightarrow q) \leftrightarrow \sim(p \lor q) \)
\( (p \rightarrow q) \leftrightarrow \sim(p \lor q) \equiv (T \rightarrow T) \leftrightarrow \sim(T \lor T) \)
\( \equiv T \leftrightarrow \sim T \)
\( \equiv T \leftrightarrow F \)
\( \equiv F \)
Hence, truth value is F.
If p and q are true and r and s are false, find the truth value of the following compound statement.
\( [(p \lor s) \rightarrow r] \lor \sim [\sim (p \rightarrow q) \lor s] \)
\( [(p \lor s) \rightarrow r] \lor \sim [\sim (p \rightarrow q) \lor s] \)
\( \equiv [(T \lor F) \rightarrow F] \lor \sim[\sim (T \rightarrow T) \lor F] \)
\( \equiv (T \rightarrow F) \lor \sim (\sim T \lor F) \)
\( \equiv F \lor \sim (F \lor F) \)
\( \equiv F \lor \sim F \)
\( \equiv F \lor T \)
\( \equiv T \)
Hence, truth value is T.
If p and q are true and r and s are false, find the truth value of the following compound statement.
\( \sim [p \lor (r \land s)] \land \sim [(r \land \sim s) \land q] \)
\( \sim [p \lor (r \land s)] \land \sim [(r \land \sim s) \land q] \)
\( \equiv \sim [T \lor (F \land F)] \land \sim [(F \land \sim F) \land T] \)
\( \equiv \sim (T \lor F) \land \sim [(F \land T) \land T] \)
\( \equiv \sim T \land \sim (F \land T) \)
\( \equiv F \land \sim F \)
\( \equiv F \land T \)
\( \equiv F \)
Hence, truth value is F.
Assuming that the following statement is true,
p : Sunday is holiday,
q : Ram does not study on holiday,
find the truth values of the following statements.
Sunday is not holiday or Ram studies on holiday.
Given p is True (T), q is True (T).
"Sunday is not holiday" is \( \sim p \).
"Ram studies on holiday" is \( \sim q \).
Symbolic form of the given statement is \( \sim p \lor \sim q \).
\( \therefore \sim p \lor \sim q \equiv \sim T \lor \sim T \)
\( \equiv F \lor F \)
\( \equiv F \)
Hence, truth value is F.
Assuming that the following statement is true,
p : Sunday is holiday,
q : Ram does not study on holiday,
find the truth values of the following statements.
If Sunday is not holiday then Ram studies on holiday.
Given p is True (T), q is True (T).
"Sunday is not holiday" is \( \sim p \).
"Ram studies on holiday" is \( \sim q \).
Symbolic form of the given statement is \( \sim p \rightarrow \sim q \).
\( \therefore \sim p \rightarrow \sim q \equiv \sim T \rightarrow \sim T \)
\( \equiv F \rightarrow F \)
\( \equiv T \)
Hence, truth value is T.
Assuming that the following statement is true,
p : Sunday is holiday,
q : Ram does not study on holiday,
find the truth values of the following statements.
Sunday is a holiday and Ram studies on holiday.
Given p is True (T), q is True (T).
"Sunday is a holiday" is p.
"Ram studies on holiday" is \( \sim q \).
Symbolic form of the given statement is \( p \land \sim q \).
\( \therefore p \land \sim q \equiv T \land \sim T \)
\( \equiv T \land F \)
\( \equiv F \)
Hence, truth value is F.
If p : He swims
q : Water is warm
Give the verbal statement for the following symbolic statement.
\( p \leftrightarrow \sim q \)
He swims if and only if water is not warm.
If p : He swims
q : Water is warm
Give the verbal statement for the following symbolic statement.
\( \sim (p \lor q) \)
It is not true that he swims or water is warm.
If p : He swims
q : Water is warm
Give the verbal statement for the following symbolic statement.
\( q \rightarrow p \)
If water is warm then he swims.
If p : He swims
q : Water is warm
Give the verbal statement for the following symbolic statement.
\( q \land \sim p \)
Water is warm and he does not swim.
