Let a be an integers Prove that a is odd if and only if ther

Let \'a\' be an integers. Prove that \'a\' is odd if and only if there is an integer \'k\' such that a= 2k1.

FIRST PROVE (P --> Q)

THEN PROVE (Q --> P)

Solution

Ques 1

for P --> Q

a = 2k-1

k can be even number also and odd number also

k = 2j+1

so putting the value of k in the equation given

a = 2(2j+1)-1

a = 4j+2-1;

a = 4j+1;

so by the defination of odd no a is odd for odd value of k

so there exists a j such that k=2j

so putting the value of k in given equation

a = 2(2j)-1;

a = 4j-1

since 4j will always be even so if we subtract 1 from even no the result will be odd no

so a is odd for even value of k

hence prove

Ques 2

For q à p

The ques is not clear which condition is to prove

Can you please elaborate