Question

1) Write down the negation of the following compound statement.

i) All rational numbers are real and complex.

2) Write down the negation of the following compound statement.

ii) All real numbers are rationals or irrational.

3) Write down the negation of the following compound statement.

iii) $x=2\text{and}x=3\text{are roots of the Quadratic equation}{x}^2-5x+6=0$

4) Write down the negation of the following compound statement.

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

5) Write down the negation of the following compound statement.

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

6) Write down the negation of the following compound statement.

vi) All prime integers are either even or odd.

7) Write down the negation of the following compound statement.

vii) $|x|$ is equal to either $xor-x.$

8) Write down the negation of the following compound statement.

viii) 6 is divisible by 2 and 3.

Answer 1

1) Given statement: ''All rational numbers are real and complex''

The component statements of given

compound statement are:

$p:$ All rational number are real

$q:$ All rational number are complex.

Hence Given statement $=\left(p\Lambda q\right)$

Now,

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

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

Therefore, negation of $\left(p\Lambda q\right)$ is given by

$\sim \left(p\Lambda q\right)=(\sim p\vee \sim q)$

= All rational numbers are not real or not complex.

2) Given statement: ''All real numbers are rational or irrationals''

The component statements of given compound statement are:

$p:$ All real numbers are rationals.

$q:$ All real numbers are irrationals.

Hence given statement $=(p\vee q)$

Now,

$\sim p:$ All real numbers are not rationals.

$\sim q:$ All real numbers are not irrationals.

Therefore, negation of $(p\vee q)$ is given by

$\sim (p\vee q)=\sim p\Lambda \sim q$

= All real numbers are not rational and all real numbers are not irrationals.

3) Given statement: ${}^{\prime \prime }x=2$ and $x=3$ are roots of the Quadratic equation $x^2-5x+6=0^{\prime \prime }$

The component statements of given compound statement are:

$p:x=2$ is a root of Quadratic

equation $x^2-5x+6=0$

$q:x=3$ is a root of Quadratic

equation $x^2-5x+6=0$

Hence given statement $=\left(p\Lambda q\right)$

$\sim p:x=2$ is not a root of Quadratic

equation $x^2-5x+6=0$

$\sim q:x=3$ is not a root of Quadratic

equation $x^2-5x+6=0$

Therefore, negation of $\left(p\Lambda q\right)$ is given by

$\sim \left(p\Lambda q\right)=\sim p\vee \sim q$

$\Rightarrow x=2$ is not a root of Quadratic equation

$x^2-5x+6=0orx=3$ is not a root of Quadratic equation $x^2-5x+6=0$

4) Given statement: ''A triangle has either 3-sides or 4-sides''

The component statements of given compound statement are:

$p:$ A triangle has 3 sides.

$q:$ A triangle has 4 sides.

Hence Given statement $=(p\vee q)$

Now,

$\sim p:$ A triangle doesnot have 3 sides.

$\sim q:$ A triangle doesnot have 4 sides.

Therefore, negation of $(p\vee q)$ is given by

$\sim (p\vee q)=\sim p\Lambda \sim q$

= A triangle has neither 3-sides nor 4-sides.

5) Given statement: ''35 is a prime number or a composite number''

The component statements of given compound statement are:

$p:35$ is a prime number.

$q:35$ is a composite number.

Hence Given statement $=(p\vee q)$

Now,

$\sim p:35$ is not a prime number.

$\sim q:35$ is not a composite number.

Therefore, negation of $(p\vee q)$ is given by

$\sim (p\vee q)=\sim p\Lambda \sim q$

= 35 is not a prime number and not a composite number.

6) Given statement: ''All prime integers are either even or odd''

The component statements of given compound statement are:

$p:$ All prime integers are even.

$q:$ All prime integers are odd.

Hence Given statement $=(p\vee q)$

Now,

$\sim p:$ All prime integers are not even.

$\sim q:$ All prime integers are not odd.

Therefore, negation of $(p\vee q)$ is given by

$\sim (p\vee q)=\sim p\Lambda \sim q$

= All prime integers are neither even nor odd.

This can also be given as :

It is false that all prime integers are either even or odd.

7) Given statement: ${}^{\prime \prime }|x|$ is equal to either $xor-x^{\prime \prime }.$

The component statements of given compound statement are:

$p:|x|$ is equal to $x.$

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

Hence Given statement $=(p\vee q)$

Now,

$\sim p:|x|$ is not equal to $x.$

$\sim q:|x|$ is not equal to $-x.$

Therefore, negation of $(p\vee q)$ is given by

$\sim (p\vee q)=\sim p\Lambda \sim q$

$=|x|$ is not equal to $x\text{and}-x$

8) Given statement: ''6 is divisible by 2 and 3''

The component statements of given compound statement are:

$p:6$ is divisible by $2.$

$q:6$ is divisible by $3.$

Hence Given statement $=\left(p\Lambda q\right)$

Now,

$\sim p:6$ is not divisible by $2.$

$\sim q:6$ is not divisible by $3.$

Therefore, negation of $\left(p\Lambda q\right)$ is given by

$\sim \left(p\Lambda q\right)=\sim p\vee \sim q$

= 6 is not divisible by 2 or 6 is not divisible by 3