Check the validity of the following statement
(a) r : if x is a real number such that `x^3+9x=0` then x is 0.
(b) r : If x is an integer and `x^3` is even , then x is also even
(c) r : If a polygon is regular then it is equiangular and equilateral.

Explanation

(a) Component statements are
p : x is a real number such that `x^3+9x=0`
q : x is 0
Now to check validity of r
(i) Direct Method : Let p is true, then q is true.
`therefore` x is real number such that `x^3+9x=0`
`therefore` x is a real number such that `x(x^2+9)=0`
`therefore x=0`
`therefore` q is true
Thus , p is true `rArr` q is true
So, given statement r is true
(ii) Contrapositive Method : Let q is false , i.e.,`x ne 0`
Now , according to p
x is real number such that `x^3+9x=0`
So, `x(x^2+9)=0`
It shows that x is real number and x=0 , but according to q , `x ne 0` , so this is a contradiction
So, r is true
Component statement are
p : x is an integer and `x^3` is even
q : x is an even integer
Suppose q is false (i.e.,~q is true) i.e.,x is an odd integer
So, `x^3= " odd"xx "odd"xx"odd "`
`=" (odd"xx "odd)"xx"odd "`
`" odd"xx "odd"`
`x^3` is also odd , so p is false
It shows that `~q rArr ~p`
(c) Component statements of r be
p : A polygon is regular.
q : A polygon is equiangular and equilateral.