Question

1) Write down the contrapositive of the following statement.

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

2) Write down the contrapositive of the following statement.

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

3) Write down the contrapositive of the following statement.

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

4) Write down the contrapositive of the following statement.

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

5) Write down the contrapositive of the following statement.

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

6) Write down the contrapositive of the following statement.

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

7) Write down the contrapositive of the following statement.

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

Answer 1

1) Given statement:

$"\text{If}x=y\text{and}y=3,\text{then}x=3"$

$\text{Let}p:x=y\text{and}y=3$

$\therefore \sim P:x\ne y\text{or}y\ne 3$

$[\because \sim (a\wedge b)=(\sim aV\sim b\left)\right]$

$q:x=3$

$\therefore \sim q:x\ne 3$

Now, Contrapositive of given statement:

$p\to q\text{is}\sim q\to \sim p$

$\text{If}x\ne 3,\text{then}x\ne y\text{or}y\ne 3$

2) Given statement: "If $n$ is a natural number, then $n$ is an integer"

Let $p:n$ is a natural number

$\therefore \sim p:n$ is not a natural number

q : n is an integer.

$\therefore \sim q:n$ is not an integer.

Now, Contrapositive of given statement:

$p\to q\text{is}q\to \sim p$

If $n$ is not an integer, then ot is not a natural number.

3) Given statement: "If all three sides of a triangle are equal, then the triangle is equilateral"

Let p: All three sides of a triangle are equal.

$\therefore \sim p:$ All three sides of a triangle are not equal.

q : The triangle is equilateral.

$\therefore \sim q:$ The triangle is not equilateral.

Now, Contrapositive of given statement:

$p\to q\text{is}\sim q\to \sim p$

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

4) Given statement: "If $x$ and $y$ are negative integers, then $xy$ is positive"

Let p : x and $y$ are negative integers.

$\therefore \sim p:$ Either $x$ or $y$ is not negative integer.

$[\because \sim (a\wedge b)=(\sim a\vee \sim b\left)\right]$

$q:xy$ is positive integer.

$\therefore \sim q:xy$ is not positive integer.

Now, Contrapositive of given statement

$p\to q\text{is}\sim q\to \sim p$

If $xy$ is not positive integer, then either $x$ or $y$ is not negative integer.

5) Given statement "If natural number $n$ is divisible by $6$, then $n$ is divisible by $2$ and $3^{\prime }$

Let p: natural number $n$ is divisible by 6

$\therefore \sim p:$ natural number $n$ is not divisible by 6.

q: natural number $n$ is divisible by $2$ and $3$.

$\therefore \sim q:$ natural number $n$ is not divisible by $2$ or $3$.

Now, Contrapositive of given statement

$p\to q\text{is}\sim q\to \sim p$

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

6) Given statement" "If it snows, then the weather will be cold"

Let p: It snows.

$\therefore \sim p:$ It does not snow.

q : The weather will be cold.

$\therefore \sim q:$ The weather will not be cold.

Now, Contrapositive of given statement

$p\to q\text{is}\sim q\to \sim p$

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

7) Given statement: "If $x$ is a real number such that $0<x<1,\text{then}{x}^2<1."$

Let p : x is a real number such that $0<x<1$

$\therefore \sim p:x$ is not a real number such that $0<x<1$

$q:x^2<1$

$\therefore \sim q:x^2>1$

Now, Contrapositive of given statement

$p\to q\text{is}\sim q\to \sim p$

If $x^2$ > 1, then $x$ is not a real number such that $0<x<1.$