### LOGIC HSC MATHS

Ex. No. 1.1

In case of statements, Write down the truth value.

1. March is the month in English Calendar.

Ans. It is a statement in Logic. Its truth value is T.
2. May God bless you!

Ans. It is an Exclamatory sentence, hence it is not a statement in Logic.
3. The sum of interior angles of a quadrilateral is 3600.

Ans. It is a statement in Logic. Its truth value is T.
4. Every natural number is a real number.

Ans. It is a statement in Logic. Its truth value is T.
5. Is the car yellow in colour?

Ans. It is an Interrogative sentence, hence it is not a statement in Logic.
6. Every quadratic equation has only real roots.

Ans. It is a statement in Logic, Its truth value is F.
7. √-4 is a rational number.

Ans. It is a statement in Logic. Its truth value is F.
8. x2 – 5x + 6 = 0 implies that x = - 2 or x = - 3
Ans. It is a statement in Logic. Its truth value is F.
9. The sum of cube roots of unit is one.

Ans. It is a statement in Logic. Its truth value is F.

Ans. It is an imperative(request) sentence, hence it is not a statement in Logic.
11. He is a bad person.

Ans. It is an open sentence, hence it is not a statement in Logic.
12. Two is the only even prime number.

Ans. It is a statement in Logic. Its truth value is T.
13. sin 3θ = 3 sinθ – 4 sin3θ ,  for all θ R.
Ans. It is a statement in Logic. Its truth value is T.
14. What a horrible sight it was!

Ans. It is an exclamatory sentence, hence it is not a statement in Logic.
15. Do not come inside the room.

Ans. It in an imperative(order) sentence, hence it is not a statement in Logic.
16. x2 + 5x + 6 = 0 , x = -2.

Ans. It is a statement in Logic. Its truth value is T.
17. Can you speak in English?

Ans. It is an Interrogative sentence, hence it is not a statement in Logic.
18. The square of every real number is positive.

Ans. It is a statement in Logic. Its truth value is F.
19. It is blue in colour.

Ans. It is an open sentence, hence it is not a statement.
20. Every rectangle is a square.

Ans. It is a statement in Logic. Its truth value is F.

1. Express the following statements in symbolic form.

(i) Banana is a fruit but potato is a vegetable. [video]

(ii) Either we play kabaddi or go for cycling. [video]

(iii) Milk is White or grass is green. [video]

(iv) The drug is effective though it has side effects. [video]

(v) Shobha stays at home, while Dhanashree and Prashant go for a movie. [Video]

2. Write the truth value of each of the following statements.

(i) √2 is a rational number or  4 - 3i is a complex number. [video]

(ii) Jupiter is a planet and Mars is a star. [video]

(iii) 5 + 3 ≠ 8 or 5 × 3 < 8 [video]

(iv) 3 × 0 = 3 and 3 + 0 = 3 [video]

(v) 16 is a perfect square but 17 is a prime number. [video]

(vi) 6 is an even number or Pune is a harbour. [video]

Ex. No. 1.3

1. Write the negation of each of the following statements: [video]

(i) π is an irrational number.

(ii) 7 is greater than 5.

(iii) Zero is not a complex number.

(iv) It is not true that mangoes are inexpensive.

2. Write the truth value of the negation of each of the following statements. [VIDEO]

(i) The sun sets in the east.

(ii) Re (z) ≤ |z| , where z is a complex number.

(iii) Moscow is in America.

iv) cos2θ + sin2θ = 1, for all θ Є R.

Ex. No. 1.4

1. Write the following statements in symbolic form. [video]

(i) I like reading but not singing.

(ii) Yuvraj neither plays cricket nor tennis.

(iii) Rama and Shama are twins.

(iv) It is not true that 'i' is a real number.

(v) Either 49 is a perfect square or 39 is divisible by 11

(vi) Pinku never works hard yet she gets good marks. [video]

(vii) Even though it is not cloudy, it is still raining. [video]

(viii) Milk is white if and only if the sky is not blue. [video]

(ix) If Kutub - minar is in Delhi then Hyderabad is in Andhra Pradesh. [video]

2. Find the truth values of the following statements.

1. SEBI office is in Mumbai or Dalal Street is in London.

Ans. Let p = SEBI office is in Mumbai.
q = Dalal Street is in London.

The truth value of p is T and q is F.

Symbolic form is

p V q = T V F = T
the truth value of the given
statement is T.

(ii) Neither 21 is a prime number nor it is divisible by 3.

Ans. Let P = 21 is a prime number
q = 21 is divisible by 3.

Here, the truth value of p is F and q is T.

The symbolic form is

~p ~ q = ~F ~T = T F = F
the truth value of the given
statement is F.

(iii) It is not true that 3 - 7i is a real number. [video]

Ans. Let p = 3 - 7i is a real number.

Here, the truth value of p is F.

The symbolic form is
~ p = ~F = T.
the truth value of the given
statement is T.

(iv) In a credit transaction, party’s name must be given and single entry system is costly. [video]

Ans. Let P = Party’s name must be given in a
credit transaction.

q = Single Entry system is costly in a credit
transaction.

Here, the truth value of p is T and q is F.

The symbolic form is
p q = T F = F
the truth value of the given
statement is F.

(v) If Joint venture is a temporary partnership, then discount on purchases is credited to supplier.

Let P = Joint venture is a temporary partnership.

q = Discount on purchases is credited to Supplier.

Here the truth value of p is T and q is F.

The symbolic form is
p -> q = T -> F = F

the truth value of the given
statement is F.

(vi) Every accountant is free to apply his own accounting rules if and only if machinery is an asset.

Ans. Let P = Every accountant is free to apply his
own accounting rules.

q = Machinery is an asset.

Here, the truth value of p is F and q is T.

The symbolic form is
p <-> q   = F <-> T = F.

the truth value of the given
statement is F.

3. If p and q are true and r and s are false statements, find the truth value of each of the following.

1. p v (q r)

Ans. Truth value of p and q are T and truth value
of r and s are F.

p v (q r) = T v (T F)
p v (q r) = T v F
p v (q r) = T

The truth value of the given statement is T

(ii) (p ~ r) (~ q s )

Ans. Truth value of p and q are T and truth value
of r and s are F.

(p ~ r) (~ q s ) = (T ~ F) (~ T F )
(p ~ r) (~ q s ) = (T T) (F F )
(p ~ r) (~ q s ) = T F
(p ~ r) (~ q s ) = F

the truth value of the given statement is F.

(iii) (p -> q) v (r <-> s)

Ans. Truth value of p and q are T and truth value
of r and s are F.

(p -> q) v (r <-> s) = (p -> q) v (r <-> s)
(p -> q) v (r <-> s) = (T -> T) v (F <-> F)
(p -> q) v (r <-> s) = T v T
(p -> q) v (r <-> s) = T

the truth value of the given statement is T.

iv. [(p -> q) -> (q -> r)] -> (r -> s)

Ans. Truth value of p and q are T and truth value
of r and s are F.

[(p -> q) -> (q -> r)] -> (r -> s) = [(T -> T) -> (T -> F)] -> (F -> F)
[(p -> q) -> (q -> r)] -> (r -> s) = (T -> F) -> T
[(p -> q) -> (q -> r)] -> (r -> s) = F -> T
[(p -> q) -> (q -> r)] -> (r -> s) = T

the truth value of the given statement is T.

v. [p v (q r) ] v ~ [(p q) v (r v s)]

Ans. Truth value of p and q are T and truth value
of r and s are F.
[pv(qr)] v ~[(pq) v (r v s)] = [Tv(TF)] v ~ [(TT) v (FvF)]
[pv(q r)] v ~ [(p q) v (r v s)] = (T v F) v ~(T v F)
[pv(q r)] v ~ [(p q) v (r v s)] = T v ~T
[pv(q r)] v ~ [(p q) v (r v s)] = T v F
[pv(q r)] v ~ [(p q) v (r v s)] = T

the truth value of the given statement is T.

4. Assuming that the following statements are true:
p: Yuvraj has sufficient money.
q : He will  buy a car.

Find the truth value of the following statements.

1. Yuvraj has sufficient money and he will buy a car.

Ans. The symbolic form of the given statement is
p q = T T = T

The truth value of the given statement is T.

1. If Yuvraj has sufficient money then he will not buy a car.

Ans. They symbolic form of the given statement is
p -> ~ q = T -> ~T = T -> F = F.

The truth value of the given statement is F.

1. Yuvraj does not have a money  or he will buy a car.

Ans. The symbolic for of the given statement is
~ p v q = = ~ T v T = F v T = T.
The truth value of the given statement is T.

5. If p = It is a day time.
q : It is warm.
Give the verbal statements for the following symbolic statements:

1. p ~ q : It is a day time and it is not warm.
2. p v q : It is a day time or it is warm.
3. p -> q : If it is a day time then it is warm.
4. q <-> p : It is warm if and only if it is a day time.

EX. NO. 1.5

1. If B = { 5, 6, 8, 10}, determine the truth value of each of the following.

1. x B, such that 3x + 4 = 28.

Ans. Clearly x = 8 B satisfies 3x + 4 = 28.
So the given statement is true.
Hence the truth value of the given statement is T.

(ii) x B, x + 7 < 14.

Ans. x = 8 B and x = 10 B do not satisfy x + 7 < 14.
So the given statement is false.
Hence the truth value of the given statement is F.

(iii) x B, 4x - 3 ≥ 17.

Ans. Clearly all the elements x B satisfy 4x - 3 ≥ 17.
So the given statement is True.
Hence the truth value of the given statement is T.

(iv) x B, such that x is an even.

Ans. Clearly, x = 6 B, x = 8 B, and x = 10 B satisfy x is an even.
So the given statement is true.
Hence the truth value of the given statement is T.

(v) y B, such that (y - 10) N.

Ans. No element y B satisfy (y - 10) N.
So the given statement is false,
Hence its truth value is F.

2. Use quantifiers to convert each of the following open sentences defined on N, into a true statement:

1.

Ans. x N, such that   is a true statement.
Note: x = 6 N satisfies

(ii) 5x - 3 < 10.

Ans. x N, such that 5x - 3 < 10 is a true statement.
Note: x = 1 N, x = 2 N satisfy 5x - 3 < 10.

(iii) x - 7 = 9.

Ans. x N, such that x - 7 = 9 is a true statement.
Note: x = 16 N satisfies x - 7 = 9.

(iv)

Ans. x N, such that   is a true statement.
Note: y = 1 N, y = 2 N satisfy  .

(v)

Ans. x N, such that   is a true statement.
Note: y = 5 N, y = 6 N satisfy

(vi)

Ans. x N, such that   is a true statement.
Note: All x N satisfy

Ex. No. 1.6

1. Prepare the truth tables of the following statement patterns:

1. (p q) -> (~p)
The truth table is given by

 p q p ⋀ q ~p (p ⋀ q) -> (~p) T T T F F T F F F T F T F T T F F F T T

(ii) (p -> q) <-> (~p v q)
 p q p -> q ~p ~p v q (p -> q) <-> (~p v q) T T T F T T T F F F F T F T T T T T F F T T T T