Please solve a and b 3 Consider the following statement Sta

Please solve a and b

3 . Consider the following statement Statement A: integers m and n, if 2m | n is odd then m and then m and n are both odd (a) Write a negation for Statement A. (b) Disprove Statement A. That is slow that Statement A is false :

Solution

Statement A: For every integers m and n, if 2m | n is odd then m and n are both odd

Negation for Statement A

Since this statement is A and B type, hence negation statement will be

For every integers m and n, if 2m | n is odd then m is not odd or n is not odd

(b)

Now we need to disprove the above statement

We can prove by two methods, first by giving a counter example

Let m = 6 and n =3

then 2m = 12 which is divisible by 3, but m is even and n is odd hence the given statement is false

Generic method

Let m be an even integer = 2p

and n be an odd integer = 2q + 1

2m = 4p

4p = k(2q+1)

k= 4p/(2q+1)

putting q=1 as odd integer, we can put p as 6 to get the answer, hence k can be integer for some value of p and q

hence the given statement A is false