Ex. No. 1.1
State which of the following sentences are statements. Justify your answer.
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 360^{0}.
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. x^{2} – 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.
10. Please, come here.
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 sin^{3}θ , 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. x^{2} + 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) cos^{2}θ + sin^{2}θ = 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.
 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.
 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(q⋀r)] v ~[(p⋀q) v (r v s)] = [Tv(T⋀F)] v ~ [(T⋀T) 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.
 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.
 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.
 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:
 p ⋀ ~ q : It is a day time and it is not warm.
 p v q : It is a day time or it is warm.
 p > q : If it is a day time then it is warm.
 q <> p : It is warm if and only if it is a day time.
EX. NO. 1.5
 If B = { 5, 6, 8, 10}, determine the truth value of each of the following.
 ∃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:

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
 Prepare the truth tables of the following statement patterns:
 (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

No comments:
Post a Comment