Check the validity of the following statement
(a) p : 77 is a multiple of 11 or 7
(b) q : If an integer x is multiple of 20 , then it is multiple of 10
(c ) For the given statement, identify the necessary and sufficient condition
r : If numerator of a ratio is smaller than denomenator then the ratio is in proper fraction

Explanation

(a) All components statements are true , so statement is valid
(b) direct Method : Let us assume that an integer x is multiple of 20 . So x can be written as
`rArr x=20n` [where n is an integer]
`rArr x=10.2n`
`rArr x=10` .m [ where m=2n is an integer]
So, x is also divisible by 10, hence another component statement is also true . So q is true
(c) Let the statement are
p : Numerator of a ratio is smaller than denominator
q : Ratio is in proper fraction
So, r is in the form "If p then q"
We know that in such case q is necessary condition for p
So , for numerator of a ratio to be smaller than senominator . It is necessary that ratio is in proper fraction
Also, p is a sufficent condition for q
Also , lnowing that numerator of a ratio is smaller than denominator is sufficent for a ratio to be in proper fraction