Prof. Bhambwani’s
RELIABLE / S.Y.J.C. / MATHS / LOGIC
EXERCISE – 1
Examine which of the following sentences are statements in logic and state their truth values.
1.(i)The earth is a planet.
(ii)Are you regular in doing your home-work?
(iii)May God bless you!
(iv)An empty vessel contains nothing.
(v)“Hum Aapke Hai Kaun” is the biggest hit of the decade.
(vi)With sunrise the night ends.
(vii)Keep your mouth shut.
(viii)Beware of dogs
(ix)He is running.
(x)`TITANIC’ is the most expensive movie ever produced by the Hollywood.
2.(i)1 is a prime number
(ii)x + 0 = x, x I.
(iii)The sum of the angles of a quadrilateral is 180.
(iv)sec2=1 + tan2 for all real such that cos ≠ 0.
(v)Every statement has one truth value
(vi)x + 3 = 10, x I.
(vii) is an irrational number.
(viii)The quadratic equation ax2 + bx + c = 0, a ≠ 0 has always two real roots.
(ix)The square of a real number is positive
(x)Given two coplanar lines, they are either parallel or intersecting.
(xi)Given two straight lines, they are either parallel or intersecting.
B
3.Represent the following statements (assuming them to be true) by Venn diagrams.
(i)No politician is honest.
(ii)Some students are hard working.
(iii)Some quadratic equations have two equal roots.
(iv)All mathematicians are introverts
(v)Sunday implies a holiday.
(vi)No quadrilateral is a triangle.
(vii)Equilateral triangles are isosceles.
(viii)Many government servants are not corrupt.
(ix)I am rich if and only if I own a bungalow at Khandala
(x)Some parallelograms are rectangles.
(xi)If a quadrilateral is a rhombus then it is a parallelogram.
(xii)Some non resident Indians are not patriotic.
(xiii)Some rational numbers are not integers.
EXERCISE – 2
A
4.Write the following statements in symbolic form:
(i)Rashmi is a scholar and Raj is a sportsman.
(ii)Akhil likes mathematics but not chemistry.
(iii)e is a vowel or 2 + 3 = 5.
(iv)If 58 = 29 X 2 then lions can drive the car
(v)(a + b)2 = a2 + b2 if ab = 0
(vi)If the question paper is not easy then we shall not pass.
(vii)Darry1 studies hard but does not get good marks in the examination.
(viii)The drug is effective though it has side effects.
5.Let P denote `price increases’ and q denote `demand falls’. Express the following statements in symbolic form.
(i)If price increases then demand falls.
(ii)If demand does not fall then price does not increase.
(iii)If price does not increase then demand does not fall.
(iv)Price increases if and only if demand falls.
6.Express the following statements in symbolic form using the statements, P: Aishwarya is tall, q: Aishwarya is beautiful.
(i)Aishwarya is tall and beautiful.
(ii)Aishwarya is tall but not beautiful.
(iii)Aishwarya is not tall but beautiful.
(iv)Aishwarya is neither tall nor beautiful
(v)If Aishwarya is not tall, then she is not beautiful.
7.Express the following statements in symbolic form using the statements, p: I am brave, q: I will climb the Mount Everest.
(i)I am brave but I will not climb the Mount Everest.
(ii)If I climb the Mount Everest then I am brave.
(iii)If I am not brave then I will not climb the Mount Everest
(iv)I am neither brave nor I will climb the Mount Everest.
(v)I am brave is a necessary and sufficient condition to climb the Mount Everest.
B
8.Let p: Ramu is hard working and q: Laxmi is intelligent.
Write a verbal statement to describe each of the following.
(i)p q (ii)p q (iii) ~p ~q (iv) ~q ~p (v) p ^ ~q.
9.Let p: A number is greater than 20 and q : A number is greater than 10.
Write a verbal statement to describe each of the following
(i) q ^ ~p (ii) p q (iii) p ~q.
10.Let p : The questions papers are easy q: We shall pass the examination
r : The principal declares a holiday s: We are happy.
Write a verbal statement to describe each of the following.
(i) p q(ii) r s
(iii)~ p ~ q(iv)(p r) s.
11.Let p: Tarun passed Engineering and q: Tarun will go to America.
Write a verbal statement to describe each of the following.
(i) p ^ q (ii) `p (iii) p q
(iv) ~ q ~p.
12.Let p: I win a lottery and q : I buy a `Audi Q60’ car.
Write a verbal statement to describe each of the following.
(i)p ^ ~ q(ii)q p
(iii)p ~ q(iv) ~ p ^ ~ q.
C
13.State the converse, inverse and contra positive of each of them
(i)If you go to the Himilayas, you will get peace of mind.
(ii)If it rains, the match will be cancelled.
(iii)If I do not secure good marks then I cannot go for engineering.
(iv)If x is rational then x is real.
(v)If x = y then x2 = y2.
(vi)If two triangles are congruent then their areas are equal.
D
14.Determine the truth values of the following statements.
(i)6 + 5 ≠ 11 or 0 > - 4
(ii)30 ÷ 10 ≠ 3 and 9 – 2 = 7
(iii)e is a vowel and 12 X 3 = 36.
(iv)Mumbai is in Maharashtra or height of the Himalayas is 100 metres.
(v)5 X 4 = 20 if and only if
8–3 ≠ 5.
(vi)If √9 = 2 then 7 > 3.
(vii)If 5 – 2 = 1 then 21 ÷ 3 = 8
(viii)11+2 ≠ 13, if and only if–5> 1.
(ix)It is not true that 3+3=6or4+ 7=10
(x)It is not true that 3 + 3 ≠ 6 and 4 + 7 = 11.
15.If p,q,r are statements with truth values, T, F, F, respectively, determine the truth values of the following. (i) p q (ii) q (p ^ r) (iii) (r p) q (iv) (r p) ~q.
16.If p,q,r, are statements with truth values F, T, F respectively, determine the truth values of the following. (i) p q r
(ii) p (qr) (iii) p ^ ~q r
(iv) (q r)p (v) (~p~q) r
(vi) (pq) ~ (p ~q)
EXERCISE 3
A
17.Construct the truth table for each of the following statement patterns:
(i)(p ^ q) ~p(ii) ~q ^ (p q) (iii) ~ (~ p ^ ~ q) (iv) p (q p)
(v)(p ^ ~q) (p q)
(vi)(~ p ~q) ^ (~ q ~p)
(vii) (p ^ ~q) (q p)
(viii)(~ p ~q) ~ (p ^ q)
(ix) (p ^ r) (p ~q)
(x) p [~(q ^ r)].
B
18.Determine whether the following statement patterns are tautologies contradictions or neither:
(i)p (p ^ ~q)(ii) p (p q) (iii) ~ (p ^ q) (~p ~q)
(iv) p ~ (p^q)
(v)(p ^q) (pq)
(vi)(pq) (q p)
(vii) [(p q) ^~q]~ p
(viii)[~ (pq)] ^ p)
(ix) [p q]^ (p ^ ~q)
(x) [(p q) ^(~p)] ^ (~q)
(xi) (p ^ q) ^ ~ (p q)
(xii) ~ p ^ q (xiii) (p q) ^ (q p)
(xiv)~ p (p ~ q) (xv) (p ~q) (xv) (p ~q) ^ p.
C
19.Show that the following pairs of statements are equivalent:
(i)p ^ q, q ^ p
(ii)p (q ^ r), (pq) ^(pr)
(iii)p q, ~pq
(iv)~p ^q, (p q) ^ ~p
(v)p q, ~(p ^~q) ^~(q ^ ~p)
(vi)~(pq)(~ p ^ q), ~p
(vii)p ^(q r), (p ^ q) (pq)(p ^ r)
23.Write the negations of the following statements (without using the phrase: “It is false that”/”It is not true that”).
(i)I go to the college or I go to a movie
(ii)5 X 2 = 10 or 5 + 2 ≠ 7
(iii)A is tall and B is fair.
(iv)7 – 3 = 4 and 7 ÷ 3 ≠ 4
(v)n is prime only if n is odd.
(vi)x is real if x is rational
(vii)If I do not score well then I cannot go for engineering
(viii)(a + b)2 = a2 + b2 if ab = 0
(ix)If it rains heavily then the lectures are cancelled but the students do not go home.
(x)Some professors are sincere in their work
(xi)All students are hard working.
(xii)Some sweets are made of kesar and dry fruits.
24.Find the negations of the following:
(i)p ^ ~q
(ii)p (q ^ r)
(iii)p ^ (q r)
(iv)p (q ^ r)
(v)(p q) ^ (q r)
25.State the dual of each of the following statements, by applying the principle of duality, and prove that both sides of the dual are equivalent.
(i)p ^ p p
(ii)p q q p
(iii)(p ^q) ^r p ^ (q ^ r)
(iv)p (q ^ r) (p q) ^ (p r)
(v)~(p ^q) ~p ~p ~q
++++++++++++
EXERCISE – 1
Examine which of the following sentences are statements in logic and state their truth values.
1.(i)The earth is a planet.
(ii)Are you regular in doing your home-work?
(iii)May God bless you!
(iv)An empty vessel contains nothing.
(v)“Hum Aapke Hai Kaun” is the biggest hit of the decade.
(vi)With sunrise the night ends.
(vii)Keep your mouth shut.
(viii)Beware of dogs
(ix)He is running.
(x)`TITANIC’ is the most expensive movie ever produced by the Hollywood.
2.(i)1 is a prime number
(ii)x + 0 = x, x I.
(iii)The sum of the angles of a quadrilateral is 180.
(iv)sec2=1 + tan2 for all real such that cos ≠ 0.
(v)Every statement has one truth value
(vi)x + 3 = 10, x I.
(vii) is an irrational number.
(viii)The quadratic equation ax2 + bx + c = 0, a ≠ 0 has always two real roots.
(ix)The square of a real number is positive
(x)Given two coplanar lines, they are either parallel or intersecting.
(xi)Given two straight lines, they are either parallel or intersecting.
B
3.Represent the following statements (assuming them to be true) by Venn diagrams.
(i)No politician is honest.
(ii)Some students are hard working.
(iii)Some quadratic equations have two equal roots.
(iv)All mathematicians are introverts
(v)Sunday implies a holiday.
(vi)No quadrilateral is a triangle.
(vii)Equilateral triangles are isosceles.
(viii)Many government servants are not corrupt.
(ix)I am rich if and only if I own a bungalow at Khandala
(x)Some parallelograms are rectangles.
(xi)If a quadrilateral is a rhombus then it is a parallelogram.
(xii)Some non resident Indians are not patriotic.
(xiii)Some rational numbers are not integers.
EXERCISE – 2
A
4.Write the following statements in symbolic form:
(i)Rashmi is a scholar and Raj is a sportsman.
(ii)Akhil likes mathematics but not chemistry.
(iii)e is a vowel or 2 + 3 = 5.
(iv)If 58 = 29 X 2 then lions can drive the car
(v)(a + b)2 = a2 + b2 if ab = 0
(vi)If the question paper is not easy then we shall not pass.
(vii)Darry1 studies hard but does not get good marks in the examination.
(viii)The drug is effective though it has side effects.
5.Let P denote `price increases’ and q denote `demand falls’. Express the following statements in symbolic form.
(i)If price increases then demand falls.
(ii)If demand does not fall then price does not increase.
(iii)If price does not increase then demand does not fall.
(iv)Price increases if and only if demand falls.
6.Express the following statements in symbolic form using the statements, P: Aishwarya is tall, q: Aishwarya is beautiful.
(i)Aishwarya is tall and beautiful.
(ii)Aishwarya is tall but not beautiful.
(iii)Aishwarya is not tall but beautiful.
(iv)Aishwarya is neither tall nor beautiful
(v)If Aishwarya is not tall, then she is not beautiful.
7.Express the following statements in symbolic form using the statements, p: I am brave, q: I will climb the Mount Everest.
(i)I am brave but I will not climb the Mount Everest.
(ii)If I climb the Mount Everest then I am brave.
(iii)If I am not brave then I will not climb the Mount Everest
(iv)I am neither brave nor I will climb the Mount Everest.
(v)I am brave is a necessary and sufficient condition to climb the Mount Everest.
B
8.Let p: Ramu is hard working and q: Laxmi is intelligent.
Write a verbal statement to describe each of the following.
(i)p q (ii)p q (iii) ~p ~q (iv) ~q ~p (v) p ^ ~q.
9.Let p: A number is greater than 20 and q : A number is greater than 10.
Write a verbal statement to describe each of the following (i) q ^ ~p (ii) p q (iii) p ~q.
10.Let p : The questions papers are easyq: We shall pass the examination
r : The principal declares a holiday s: We are happy.
Write a verbal statement to describe each of the following.
(i) p q(ii) r s (iii)~ p ~ q(iv)(p r) s.
11.Let p: Tarun passed Engineering and q: Tarun will go to America.
Write a verbal statement to describe each of the following.
(i) p ^ q (ii) `p (iii) p q (iv) ~ q ~p.
12.Let p: I win a lottery and q : I buy a `Audi Q60’ car.
Write a verbal statement to describe each of the following.
(i)p ^ ~ q(ii)q p(iii)p ~ q(iv) ~ p ^ ~ q.
C
13.State the converse, inverse and contra positive of each of them
(i)If you go to the Himilayas, you will get peace of mind.
(ii)If it rains, the match will be cancelled.
(iii)If I do not secure good marks then I cannot go for engineering.
(iv)If x is rational then x is real.
(v)If x = y then x2 = y2.
(vi)If two triangles are congruent then their areas are equal.
D
14.Determine the truth values of the following statements.
(i)6 + 5 ≠ 11 or 0 > - 4
(ii)30 ÷ 10 ≠ 3 and 9 – 2 = 7
(iii)e is a vowel and 12 X 3 = 36.
(iv)Mumbai is in Maharashtra or height of the Himalayas is 100 metres.
(v)5 X 4 = 20 if and only if 8 – 3 ≠ 5.
(vi)If √9 = 2 then 7 > 3.
(vii)If 5 – 2 = 1 then 21 ÷ 3 = 8
(viii)11 + 2 ≠ 13, if and only if – 5 > 1.
(ix)It is not true that 3 + 3 = 6 or 4 + 7 = 10
(x)It is not true that 3 + 3 ≠ 6 and 4 + 7 = 11.
15.If p,q,r are statements with truth values, T, F, F, respectively, determine the truth values of the following. (i) p q (ii) q (p ^ r) (iii) (r p) q (iv) (r p) ~q.
16.If p,q,r, are statements with truth values F, T, F respectively, determine the truth values of the following. (i) p q r (ii) p (qr) (iii) p ^ ~q r (iv) (q r)p (v) (~p~q) r
(vi) (pq) ~ (p ~q)
EXERCISE 3
A
17.Construct the truth table for each of the following statement patterns:
(i)(p ^ q) ~p(ii) ~q ^ (p q)(iii)~ (~ p ^ ~ q) (iv) p (q p)
(v)(p ^ ~q) (p q)(vi)(~ p ~q) ^ (~ q ~p) (vii) (p ^ ~q) (q p)
(viii)(~ p ~q) ~ (p ^ q)(ix) (p ^ r) (p ~q)(x) p [~(q ^ r)].
B
18.Determine whether the following statement patterns are tautologies contradictions or neither:
(i)p (p ^ ~q)(ii) p (p q) (iii)~ (p ^ q) (~p ~q) (iv) p ~ (p ^ q)
(v)(p ^ q) (pq) (vi)(pq) (q p) (vii) [(p q) ^ ~q] ~ p
(viii)[~ (p q)] ^ p) (ix) [p q] ^ (p ^ ~q) (x) [(p q) ^ (~p)] ^ (~ q)
(xi) (p ^ q) ^ ~ (p q) (xii) ~ p ^ q (xiii) (p q) ^ (q p)
(xiv)~ p (p ~ q) (xv) (p ~q) (xv) (p ~q) ^ p.
C
19.Show that the following pairs of statements are equivalent:
(i)p ^ q, q ^ p(ii)p (q ^ r), (pq) ^(pr)
(iii)p q, ~pq(iv)~p ^q, (p q) ^ ~p
(v)p q, ~(p ^~q) ^~(q ^ ~p)(vi)~(pq)(~ p ^ q), ~p
(vii)p ^(q r), (p ^ q) (pq)(p ^ r)
23.Write the negations of the following statements (without using the phrase: “It is false that”/”It is not true that”).
(i)I go to the college or I go to a movie
(ii)5 X 2 = 10 or 5 + 2 ≠ 7
(iii)A is tall and B is fair.
(iv)7 – 3 = 4 and 7 ÷ 3 ≠ 4
(v)n is prime only if n is odd.
(vi)x is real if x is rational
(vii)If I do not score well then I cannot go for engineering
(viii)(a + b)2 = a2 + b2 if ab = 0
(ix)If it rains heavily then the lectures are cancelled but the students do not go home.
(x)Some professors are sincere in their work
(xi)All students are hard working.
(xii)Some sweets are made of kesar and dry fruits.
24.Find the negations of the following:
(i)p ^ ~q
(ii)p (q ^ r)
(iii)p ^ (q r)
(iv)p (q ^ r)
(v)(p q) ^ (q r)
25.State the dual of each of the following statements, by applying the principle of duality, and prove that both sides of the dual are equivalent.
(i)p ^ p p
(ii)p q q p
(iii)(p ^q) ^r p ^ (q ^ r)
(iv)p (q ^ r) (p q) ^ (p r)
(v)~(p ^q) ~p ~p ~q
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