## Chapter 14 Mathematical Reasoning

### Short Answer Type Questions

**1.** Which of the following sentences are statements? Justify

(i) A triangle has three sides.

(ii) 0 is a complex number.

(iii) Sky is red.

(iv) Every set is an infinite set.

(v) $15+8>23$.

(vi) $y+9=7$

(vii) Where is your bag?

(viii) Every square is a rectangle.

(ix) Sum of opposite angles of a cyclic quadrilateral is $180^{\circ}$.

(x) $\sin ^{2} x+\cos ^{2} x=0$

## Show Answer

**Solution**

As we know, a statement is a sentence which is either true or false but not both simultaneously.

(i) It is true statement.

(ii) It is true statement.

(iii) It is false statement.

(iv) It is false statement.

(v) It is false statement.

(vi) $y+9=7$

It is not considered as a statement, since the value of $y$ is not given.

(vii) It is a question, so it is not a statement.

(viii) It is a true statement.

(ix) It is true statement.

(x) It is false statement.

**2.** Find the component statements of the following compound statements.

(i) Number 7 is prime and odd.

(ii) Chennai is in India and is the capital of Tamil Nadu.

(iii) The number 100 is divisible by 3,11 and 5.

(iv) Chandigarh is the capital of Haryana and UP.

(v) $\sqrt{7}$ is a rational number or an irrational number.

(vi) 0 is less than every positive integer and every negative integer.

(vii) Plants use sunlight, water and carbon dioxide for photosynthesis.

(viii) Two lines in a plane either intersect at one point or they are parallel.

(ix) A rectangle is a quadrilateral or a 5 sided polygon.

## Show Answer

**Solution**

(i) $p$ : Number 7 is prime.

$q$ : Number 7 is odd.

(ii) $P$ : Chennai is in India.

$q$ : Chennai is capital of Tamil Nadu.

(iii) $p: 100$ is divisible by 3 .

$q: 100$ is divisible by 11 .

$r: 100$ is divisible by 5 .

(iv) $p$ : Chandigarh is capital of Haryana.

$q$ : Chandigarh is capital of UP.

(v) $p: \sqrt{7}$ is a rational number.

$q: \sqrt{7}$ is an irrational number.

(vi) $p: 0$ is less than every positive integer.

$q: 0$ is less than every negative integer.

(vii) $p$ : Plants use sunlight for photosysthesis.

$q$ : Plants use water for photosynthesis.

$r$ : Plants use carbon dioxide for photosysthesis.

(viii) $p$ : Two lines in a plane intersect at one point.

$q$ : Two lines in a plane are parallel.

(ix) $p:$ A rectangli, is a quadrilateral.

$q$ : A rectangle is a 5 -sided polygon.

**3.** Write the component statements of the following compound statements and check whether the compound statement is true or false.

(i) 57 is divisible by 2 or 3.

(ii) 24 is a multiple of 4 and 6.

(iii) All living things have two eyes and two legs.

(iv) 2 is an even number and a prime number.

## Show Answer

**Solution**

(i) Given compound statement is of the form ’ $p v q$ ‘. Since, the statement ’ $p v q$ ’ has the truth value $T$ whenever either $p$ or q or both have the truth value $T$.

So, it is true statement.

Its component statements are

$p: 57$ is divisible by $2 . \quad$ [false]

$q: 57$ is divisible by $3 . \quad$ [true]

(ii) Given compound statement is of the form ’ $p \wedge q$ ‘. Since, the statement ’ $p \wedge q$ ’ have the truth value $T$ whenever both $p$ and $q$ have the truth value $T$.

So, it is a true statement.

Its component statements are

$p: 24$ is multiple of 4 [true]

$q: 24$ is multiple of $6 . \quad$ [true]

(iii) It is a false statement. Since ’ $p \wedge q$ ’ has truth value $F$ whenever either $p$ or $q$ or both have the truth value $F$.

Its component statements are

$p$ : All living things have two eyes. [false]

$q$ : All living things have two legs. [false]

(iv) It is a true statement.

Its component statements are

$p: 2$ is an even number. [true]

$q: 2$ is a prime number. [true]

**4.** Write the negative on the following simple statements.

(i) The number 17 is prime.

(ii) $2+7=6$.

(iii) Violets are blue.

(iv) $\sqrt{5}$ is a rational number.

(v) 2 is not a prime number.

(vi) Every real number is an irrational number.

(vii) Cow has four legs.

(viii) A leap year has 366 days.

(ix) All similar triangles are congruent.

(x) Area of a circle is same as the perimeter of the circle.

## Show Answer

**Solution**

(i) The number 17 is not prime.

(ii) $2+7 \neq 6$.

(iii) Violets are not blue.

(iv) $\sqrt{5}$ is not a rational number.

(v) 2 is a prime number.

(vi) Every real number is not an irrational number.

(vii) Cow has not four legs.

(viii) A leap year has not 366 days.

(ix) There exist similar triangles which are not congruent.

(x) Area of a circle is not same as the perimeter of the circle.

**5.** Translate the following statements into symbolic form

(i) Rahul passed in Hindi and English.

(ii) $x$ and $y$ are even integers.

(iii) 2, 3 and 6 are factors of 12.

(iv) Either $x$ or $x+1$ is an odd integer.

(v) A number is either divisible by 2 or 3.

(vi) Either $x=2$ or $x=3$ is a root of $3 x^{2}-x-10=0$.

(vii) Students can take Hindi or English as an optional paper.

## Show Answer

**Solution**

(i) $p$ : Rahul passed in Hindi.

$q$ : Rahul passed in English.

$p \wedge q$ : Rahul passed in Hindi and English.

(ii) $p: x$ is even integers.

$q: y$ is even integers.

$p \cap q: x$ and $y$ are even integers.

(iii) $p: 2$ is factor of 12 .

$q: 3$ is factor of 12.

$r: 6$ is factor of 12.

$p \wedge q \wedge r: 2,3$ and 6 are factor of 12 .

(iv) $p: x$ is an odd integer.

$q:(x+1)$ is an odd integer.

$p \vee q$ : Either $x$ or $(x+1)$ is an odd integer.

(v) $p:$ A number is divisible by 2 .

$q$ : A number is divisible by 3.

$p \vee q$ : A number is either divisible by 2 or 3 .

(vi) $p: x=2$ is a root of $3 x^{2}-x-10=0$.

$q: x=3$ is a root of $3 x^{2}-x-10=0$.

$p \vee q$ : Either $x=2$ or $x=3$ is a root of $3 x^{2}-x-10=0$.

(vii) $p$ : Students can take Hindi as an optional paper.

$q$ : Students can take English as an optional subject.

$p \vee q$ : Students can take Hindi or English as an optional paper.

**6.** Write down the negation of following compound statements.

(i) All rational numbers are real and complex.

(ii) All real numbers are rationals or irrationals.

(iii) $x=2$ and $x=3$ are roots of the quadratic equation $x^{2}-5 x+6=0$.

(iv) A triangle has either 3 -sides or 4 -sides.

(v) 35 is a prime number or a composite number.

(vi) All prime integers are either even or odd.

(vii) $|x|$ is equal to either $x$ or $-x$.

(viii) 6 is divisible by 2 and 3 .

## Show Answer

**Thinking Process**

Use (i) $\sim(p \wedge q)=\sim p \mathbf{V} \sim q$

(ii) $\sim(p \vee q)=\sim p \wedge \sim q$

**Solution**

(i) Let $p:$ All rational numbers are real.

$q$ : All rational numbers are complex.

$\sim p$ : All rational number are not real.

$\sim q$ : All rational numbers are not complex.

$\sim(p \wedge q)$ : All rational numbers are not real or not complex. $\quad[\because \sim(p \wedge q)=\sim p \vee \sim q]$

(ii) Let $p:$ All real numbers are rationals.

$q$ : All real numbers are irrational.

Then, the negation of the above statement is given by

$\sim(p \vee q)$ : All real numbers are not rational and all real numbers are not irrational.

$[\because \sim(p \vee q)=\sim p \wedge \sim q]$

(iii) Let $p: x=2$ is root of quadratic equation $x^{2}-5 x+6=0$.

$q: x=3$ is root of quadratic equation $x^{2}-5 x+6=0$.

Then, the negation of conjunction of above statement is given by

$\sim(p \wedge q): x=2$ is not a root of quadratic equation $x^{2}-5 x+6=0$ or $x=3$ is not a root of the quadratic equation $x^{2}-5 x+6=0$.

(iv) Let $p:$ A triangle has 3 -sides.

$q$ : A triangle has 4 -sides.

Then, negation of disjunction of the above statement is given by

$\sim(p \vee q)$ : A triangle has neither 3 -sides nor 4 -sides.

(v) Let $p: 35$ is a prime number.

$q: 35$ is a composite number.

Then, negation of disjunction of the above statement is given by

$\sim(p \vee q): 35$ is not a prime number and it is not a composite number.

(vi) Let $p:$ All prime integers are even.

$q:$ All prime integers are odd.

Then negation of disjunction of the above statement is given by

$\sim(p \vee q)$ : All prime integers are not even and all prime integers are not odd.

(vii) Let $p:|x|$ is equal to $x$.

$q:|x|$ is equal to $-x$.

Then negation of disjunction of the above statement is given by

$\sim(p \vee q):|x|$ is not equal to $x$ and it is not equal to $-x$.

(viii) Let $p: 6$ is divisible by 2 .

$q: 6$ is divisible by 3 .

Then, negation of conjunction of above statement is given by

$\sim(p \wedge q)$ : 6 is not divisible by 2 or it is not divisible by 3

**7.** Rewrite each of the following statements in the form of conditional statements.

(i) The square of an odd number is odd.

(ii) You will get a sweet dish after the dinner.

(iii) You will fail, if you will not study.

(iv) The unit digit of an integer is 0 or 5 , if it is divisible by 5.

(v) The square of a prime number is not prime.

(vi) $2 b=a+c$, if $a, b$ and $c$ are in AP.

## Show Answer

**Solution**

We know that, some of the common expressions of conditional statement $p \rightarrow q$ are

(i) if $p$, then $q$

(ii) $q$ if $p$

(iii) $p$ only if $q$

(iv) $p$ is sufficient for $q$

(v) $q$ is necesary for $p$

(vi) $\sim q$ implies $\sim p$

So, use above information to get the answer

(i) If the number is odd number, then its square is odd number.

(ii) If you take the dinner, then you will get sweet dish.

(iii) If you will not study, then you will fail.

(iv) If an integer is divisible by 5 , then its unit digits are 0 or 5 .

(v) If the number is prime, then its square is not prime.

(vi) If $a, b$ and $c$ are in $A P$, then $2 b=a+c$.

**8.** Form the biconditional statement $p \rightarrow q$, where

(i) $p$ : The unit digits of an integer is zero.

$q:$ It is divisible by 5 .

(ii) $p:$ A natural number $n$ is odd.

$q$ : Natural number $n$ is not divisible by 2 .

(iii) $p:$ A triangle is an equilateral triangle.

$q$ : All three sides of a triangle are equal.

## Show Answer

**Solution**

(i) $p \leftrightarrow q$ : The unit digit of on integer is zero, if and only if it is divisible by 5 .

(ii) $p \leftrightarrow q$ : A natural number no odd if and only if it is not divisible by 2 .

(iii) $p \leftrightarrow q$ : A triangle is an equilateral triangle if and only if all three sides of triangle are equal.

**9.** Write down the contrapositive of the following statements.

(i) If $x=y$ and $y=3$, then $x=3$.

(ii) If $n$ is a natural number, then $n$ is an integer.

(iii) If all three sides of a triangle are equal, then the triangle is equilateral.

(iv) If $x$ and $y$ are negative integers, then $x y$ is positive.

(v) If natural number $n$ is divisible by 6 , then $n$ is divisible by 2 and 3 .

(vi) If it snows, then the weather will be cold.

(vii) If $x$ is a real number such that $0<x<1$, then $x^{2}<1$.

## Show Answer

**Thinking Process**

We know that, the statement $(\sim q) \rightarrow(\sim p)$ is called contrapositive of the statement $p \rightarrow q$.

**Solution**

(i) If $x \neq 3$, then $x \neq y$ or $y \neq 3$.

(ii) If $n$ is not an integer, then it is not a natural number.

(iii) If the triangle is not equilateral, then all three sides of the triangle are not equal.

(iv) If $x y$ is not positive integer, then either $x$ or $y$ is not negative integer.

(v) If natural number $n$ is not divisible by 2 or 3 , then $n$ is not divisible by 6 .

(vi) The weather will not be cold, if it does not snow.

(vii) If $x^{2} \nless 1$, then $x$ is not a real number such that $0<x<1$.

**10.** Write down the converse of following statements.

(i) If a rectangle ’ $R$ ’ is a square, then $R$ is a rhombus.

(ii) If today is Monday, then tomorrow is Tuesday.

(iii) If you go to Agra, then you must visit Taj Mahal.

(iv) If sum of squares of two sides of a triangle is equal to the square of third side of a triangle, then the triangle is right angled.

(v) If all three angles of a triangle are equal, then the triangle is equilateral.

(vi) If $x: y=3: 2$, then $2 x=3 y$.

(vii) If $S$ is a cyclic quadrilateral, then the opposite angles of $S$ are supplementary.

(viii) If $x$ is zero, then $x$ is neither positive nor negative.

(ix) If two triangles are similar, then the ratio of their corresponding sides are equal.

## Show Answer

**Thinking Process**

We know that, the converse of the statement " $p arrow q$ " is " $(q) arrow(p)$ “.

**Solution**

(i) If thes rectangle ’ $R$ ’ is rhombus, then it is square.

(ii) If tomorrow is Tuesday, then today is Monday.

(iii) If you must visit Taj Mahal, you go to Agra.

(iv) If the triangle is right angle, then sum of squares of two sides of a triangle is equal to the square of third side.

(v) If the triangle is equilateral, then all three angles of triangle are equal. (vi) If $2 x=3 y$, then $x: y=3: 2$

(vii) If the opposite angles of a quadrilateral are supplementary, then $S$ is cyclic.

(viii) If $x$ is neither positive nor negative, then $x$ is 0 .

(ix) If the ratio of corresponding sides of two triangles are equal, then triangles are similar.

**11.** Identify the quantifiers in the following statements.

(i) There exists a triangle which is not equilateral.

(ii) For all real numbers $x$ and $y, x y=y x$.

(iii) There exists a real number which is not a rational number.

(iv) For every natural number $x, x+1$ is also a natural number.

(v) For all real numbers $x$ with $x>3, x^{2}$ is greater than 9 .

(vi) There exists a triangle which is not an isosceles triangle.

(vii) For all negative integers $x, x^{3}$ is also a negative integers.

(viii) There exists a statement in above statements which is not true.

(ix) There exists an even prime number other than 2.

(x) There exists a real number $x$ such that $x^{2}+1=0$.

## Show Answer

**Solution**

Quantifier are the phrases like ‘There exist’ and ‘For every’, ‘For all’ etc.

(i) There exists

(ii) For all

(iii) There exists

(iv) For every

(v) For all

(vi) There exists

(vii) For all

(viii) There exists

(ix) There exists

(x) There exists

**12.** Prove by direct method that for any integer ’ $n$ ‘, $n^{3}-n$ is always even.

## Show Answer

**Thinking Process**

We know that, in direct method to show a statement, if $p$ then $q$ is true, we assume $p$ is true and show $q$ is true i.e., $p arrow q$.

**Solution**

Here, two cases arise

Case I When $n$ is even,

Let

$ \begin{aligned} n & =2 K, K \in N \\ \Rightarrow\quad n^3-n & =(2 K)^{3}-(2 K)=2 K(4 K^{2}-1) \\ & =2 \lambda \text {, where } \lambda=K(4 K^{2}-1) \end{aligned} $

Thus, $(n^{3}-n)$ is even when $n$ is even.

Case II When $n$ is odd,

Let

$ \begin{aligned} n & =2 K+1, K \in N \\ \Rightarrow \quad n^{3}-n & =(2 K+1)^{3}-(2 K+1) \\ & =(2 K+1)[(2 K+1)^{2}-1] \\ & =(2 K+1)[4 K^{2}+1+4 K-1] \\ & =(2 K+1)(4 K^{2}+4 K) \\ & =4 K(2 K+1)(K+1) \\ & =2 \mu, \text { when } \mu=2 K(K+1)(2 K+1) \end{aligned} $

Then, $n^{3}-n$ is even when $n$ is odd

So, $n^{3}-n$ is always even.

**13.** Check validity of the following statement.

(i) $p: 125$ is divisible by 5 and 7.

(ii) $q: 131$ is a multiple of 3 or 11 .

## Show Answer

**Solution**

(i) $p: 125$ is divisible by 5 and 7 .

Let $q$ : 125 is divisible by 5 .

$r: 125$ is divisible by 7 .

$q$ is true, $r$ is false.

$\Rightarrow q \wedge r$ is false.

[since, $p \wedge q$ has the truth value $F$ (false) whenever either $p$ or $q$ or both have the truth value]

Hence, $p$ is not valid.

(ii) $p: 131$ is a multiple of 3 or 11 .

Let $q: 131$ is multiple of 3.

$r: 131$ is a multiple of 11.

$p$ is true, $r$ is false.

$\Rightarrow \quad p \vee r$ is true.

[since, $p \vee q$ has the truth value $T$ (true) whenever either $p$ or $q$ or both have the truth value T]

Hence, $q$ is valid.

**14.** Prove the following statement by contradiction method $p$ : The sum of an irrational number and a rational number is irrational.

## Show Answer

**Solution**

Let $p$ is false i.e., sum of an irrational and a rational number is rational.

Let $\sqrt{m}$ is irrational and $n$ is rational number.

$\begin{matrix} \Rightarrow & \sqrt{m}+n=r & \text { [rational] } \\ \Rightarrow & \sqrt{m}=r-n & \end{matrix} $

$\sqrt{m}$ is irrational, where as $(r-n)$ is rational. This is contradiction.

Then, our supposition is wrong.

Hence, $p$ is true.

**15.** Prove by direct method that for any real number $x, y$ if $x=y$, then $x^{2}=y^{2}$.

## Show Answer

**Thinking Process**

In direct method assume $p$ is true and show q is true i.e., $p \Rightarrow q$.

**Solution**

Let $p: x=y, \quad x, y \in R$

On squaring both sides,

Hence, we have the result.

$ \begin{gathered} x^{2}=y^{2}: q \\ p \Rightarrow q \end{gathered} $

**16.** Using contrapositive method prove that, if $n^{2}$ is an even integer, then $n$ is also an even integer.

## Show Answer

**Thinking Process**

In contrapositive method assume $\sim q$ is true and show $\sim$ p is true i.e., $\sim q \Rightarrow \sim p$.

**Solution**

Let $p: n^{2}$ is an even integer.

$q: n$ is also an even integer.

Let $\sim p$ is true i.e., $n$ is not an even integer.

$\Rightarrow n^{2}$ is not an even integer.

[since, square of an odd integer is odd]

$\Rightarrow \sim p$ is true.

Therefore, $\sim q$ is true $\Rightarrow \sim p$ is true.

Hence proved.

### Objective Type Questions

**17.** Which of the following is a statement?

(a) $x$ is a real number

(b) Switch off the fan

(c) 6 is a natural number

(d) Let me go

## Show Answer

**Solution**

(c) As we know a statement is a sentence which is either true or false.

So, 6 is a natural number, which is true.

Hence, it is a statement.

**18.** Which of the following is not a statement.

(a) Smoking is injurious to health

(b) $2+2=4$

(c) 2 is the only even prime number

(d) Come here

## Show Answer

**Solution**

(d) ‘Come here’ is not a statement. Since, no sentence can be called a statement, if it is an order.

**19.** The connective in the statement ’ $2+7>9$ or $2+7<9^{\prime}$ is

(a) and

(b)or

(c) $>$

$($ d $)<$

## Show Answer

**Solution**

(b) In ’ $2+7>9$ or $2+7<9$ ‘, or is the connective.

**20.** The connective in the statement “Earth revolves round the Sun and Moon is a satellite of earth” is

(a) or

(b) Earth

(c) Sun

(d) and

## Show Answer

**Solution**

(d) Connective word is ‘and’.

**21.** The negation of the statement “A circle is an ellipse” is

(a) An ellipse is a circle

(b) An ellipse is not a circle

(c) A circle is not an ellipse

(d) A circle is an ellipse

## Show Answer

**Solution**

(c) Let $p$ : A circle is an ellipse.

$\sim p$ : A circle is not an ellipse.

**22.** The negation of the statement " 7 is greater than 8 " is

(a) 7 is equal to 8

(b) 7 is not greater than 8

(c) 8 is less than 7

(d) None of these

## Show Answer

**Solution**

(b) Let $p: 7$ is greater than 8 .

$\sim p: 7$ is not greater than 8 .

**23.** The negation of the statement " 72 is divisible by 2 and 3 " is

(a) 72 is not divisible by 2 or 72 is not divisible by 3

(b) 72 is not divisible by 2 and 72 is not divisible by 3

(c) 72 is divisible by 2 and 72 is not divisible by 3

(d) 72 is not divisible by 2 and 72 is divisible by 3

## Show Answer

**Solution**

(b) Let $p: 72$ is divisible by 2 and 3 .

Let $q: 72$ is divisible by 2 .

$r: 72$ is divisible by 3 .

$\sim q: 72$ is not divisible by 2 .

$\sim r: 72$ is not divisible by 3 .

$\sim(q \wedge r): \sim q v \sim r$

$\Rightarrow 72$ is not divisible by 2 or 72 is not divisible by 3 .

**24.** The negation of the statement “Plants take in $CO_2$ and give out $O_2$ " is

(a) Plants do not take in $CO_2$ and do not given out $O_2$

(b) Plants do not take in $CO_2$ or do not give out $O_2$

(c) Plants take is $CO_2$ and do not give out $O_2$

(d) Plants take in $CO_2$ or do not give out $O_2$

## Show Answer

**Solution**

(b) Let $p$ : Plants take in $CO_2$ and give out $O_2$.

Let $q$ : Plants take in $CO_2$.

$r$ : Plants give out $O_2$.

$\sim q$ : Plants do not take in $CO_2$.

$\sim r$ : Plants do not give out $O_2$.

$\sim(q \wedge r)$ : Plants do not take in $CO_2$ or do not give out $O_2$.

**25.** The negative of the statement “Rajesh or Rajni lived in Bangaluru” is

(a) Rajesh did not live in Bengaluru or Rajni lives in Bengaluru

(b) Rajesh lives in Bengaluru and Rajni did not live in Bengaluru

(c) Rajesh did not live in Bengaluru and Rajni did not live in Bengaluru

(d) Rajesh did not live in Bengaluru or Rajni did not live in Bengaluru

## Show Answer

**Solution**

(c) Let $p$ : Rajesh or Rajni lived in Bengaluru.

and $q$ : Rajesh lived in Bengaluru.

$r$ : Rajni lived in Bengaluru.

$\sim q$ : Rajesh did not live in Bengaluru.

$\sim r$ : Rajni did not live in Bengaluru.

$\sim(q \vee r)$ : Rajesh did not live in Bengaluru and Rajni did not live in Bengaluru.

**26.** The negation of the statement " 101 is not a multiple of 3 " is

(a) 101 is a multiple of 3

(b) 101 is a multiple of 2

(c) 101 is an odd number

(d) 101 is an even number

## Show Answer

**Solution**

(a) Let $p$ : 101 is not a multiple of 3.

$\sim p: 101$ is a multiple of 3.

**27.** The contrapositive of the statement “If 7 is greater than 5 , then 8 is greater than 6 " is

(a) If 8 is greater than 6 , then 7 is greater than 5

(b) If 8 is not greater than 6 , then 7 is greater than 5

(c) If 8 is not greater than 6 , then 7 is not greater than 5

(d) If 8 is greater than 6 , then 7 is not greater than 5

## Show Answer

**Solution**

(c) Let $p: 7$ is greater than 5 .

and $q: 8$ is greater than 6 .

$\therefore p \rightarrow q$

$\sim p: 7$ is not greater than 5 .

$\sim q: 8$ is not greater than 6 .

$(\sim q) \rightarrow(\sim p)$ i.e., If 8 is not greater than 6 , then 7 is not greater than 5 .

**28.** The converse of the statement “If $x>y$, then $x+a>y+a$ " is

(a) If $x<y$, then $x+a<y+a$

(b) If $x+a>y+a$, then $x>y$

(c) If $x<y$, then $x+a<y+a$

(d) If $x>y$, then $x+a<y+a$

## Show Answer

**Solution**

(b) Let

$ \begin{gathered} p: x>y \\ q: x+a>y+a \\ p \rightarrow q \end{gathered} $

Converse of the above statement is

$ q \rightarrow p $

i.e., If $x+a>y+a$, then $x>y$.

**29.** The converse of the statement “If sun is not shining, then sky is filled with clouds” is

(a) If sky is filled with clouds, then the Sun is not shining

(b) If Sun is shining, then sky is filled with clouds

(c) If sky is clear, then Sun is shining

(d) If Sun is not shining, then sky is not filled with clouds

## Show Answer

**Solution**

(a) Let $p$ : Sun is not shining.

and $q$ : Sky is filled with clouds.

Converse of the above statement $p \rightarrow q$ is $q \rightarrow p$.

If sky is filled with clouds, then the Sun is not shining.

**30.** The contrapositive of the statement “If $p$, then $q$ “, is

(a) if $q$, then $p$

(b) if $p$, then $\sim q$

(c) if $\sim q$. then $\sim p$

(d) if $\sim p$, then $\sim q$

## Show Answer

**Solution**

(c) $p \rightarrow q$

If $p$, then $q$

Contrapositive of the statement $p \rightarrow q$ is $(\sim q) \rightarrow(\sim p)$.

If $\sim q$, then $\sim p$.

**31.** The statement “If $x^{2}$ is not even, then $x$ is not even” is converse of the statement

(a) If $x^{2}$ is odd, then $x$ is even

(b) If $x$ is not even, then $x^{2}$ is not even

(c) If $x$ is even, then $x^{2}$ is even

(d) If $x$ is odd, then $x^{2}$ is even

## Show Answer

**Solution**

(b) Let $p: x^{2}$ is not even.

and $q: x$ is not even.

Converse of the statement $p \rightarrow q$ is $q \rightarrow p$.

i.e., If $x$ is not even, then $x^{2}$ is not even.

**32.** The contrapositive of statement ‘If Chandigarh is capital of Punjab, then Chandigarh is in India’ is

(a) if Chandigarh is not in India, then Chandigarh is not the capital of Punjab

(b) if Chandigarh is in India, then Chandigarh is Capital of Punjab

(c) if Chandigarh is not capital of Punjab, then Chandigarh is not capital of India

(d) if Chandigarh is capital of Punjab, then Chandigarh is not is India

## Show Answer

**Solution**

(a) Let $p$ : Chandigarh is capital of Punjab.

and $q$ : Chandigarh is in India.

$\sim p$ : Chandigarh is not capital of Punjab.

$\sim q$ : Chandigarh is not in India.

Contrapositive of the statement $p \rightarrow q$ is

if $(\sim q)$, then $(\sim p)$.

If Chandigarh is not in India, then Chandigarh is not the capital of Punjab.

**33.** Which of the following is the conditional $p arrow q$ ?

(a) $q$ is sufficient for $p$

(b) $p$ is necessary for $q$

(c) $p$ only if $q$

(d) if $q$ then $p$

## Show Answer

**Solution**

(c) ’ $p \rightarrow q$ ’ is same as ’ $p$ only if $q$ ‘.

**34.** The negation of the statement “The product of 3 and 4 is 9 " is

(a) it is false that the product of 3 and 4 is 9

(b) the product of 3 and 4 is 12

(c) the product of 3 and 4 is not 12

(d) it is false that the product of 3 and 4 is not 9

## Show Answer

**Solution**

(a) The negation of the above statement is ‘It is false that the product of 3 and 4 is 9 ‘.

**35.** Which of the following is not a negation of “A nature number is greater than zero”

(a) A natural number is not greater than zero

(b) It is false that a natural number is greater than zero

(c) It is false that a natural number is not greater than zero

(d) None of the above

## Show Answer

**Solution**

(c) The false negation of the given statement is “It is false that a natural number is not greater than zero”.

**36.** Which of the following statement is a conjunction?

(a) Ram and Shyam are friends

(b) Both Ram and Shyam are tall

(c) Both Ram and Shyam are enemies

(d) None of the above

## Show Answer

**Solution**

(d) If two simple statements $p$ and $q$ are connected by the word ‘and’, then the resulting compound statement $p$ and $q$ is called a conjuction of $p$ and $q$ Here, none of the given statement is conjunction

**37.** State whether the following sentences are statements or not

(i) The angles opposite to equal sides of a triangle are equal.

(ii) The moon is a satellites of Earth.

(iii) May God bless you.

(iv) Asia is a continent.

(v) How are you?

## Show Answer

**Solution**

(i) It is a statement.

(ii) It is a statement.

(iii) It is not a statement, since it is an exclamations.

(iv) It is a statement.

(v) It is not a statement, since it is a questions.