Prove that the product xy is odd if and only if both x and y are odd integers .

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Let us suppose,

p = x and y are odd

q = xy is odd

i.e. p → q

First Case: p → q (direct proof)

If x and y are odd then xy is odd. By the definition of odd integers, Ǝ integers a, b.

x = 2a + 1   and  y = 2b + 1

xy = (2a + 1) (2b + 1)

= 4ab + 2a + 2b + 1

= 2(2ab + a + b) + 1

= 2c + 1,   for some integer c

Here, x is also odd by direct proof

Second Case: ¬ q → ¬ p  (indirect proof)

The product xy is odd if and only if x and y are odd.

p = product xy is odd

q = x and y are odd

Let us suppose, x and y are even

Now, We prove: ¬ q → ¬ p

By definition of even integers Ǝ integers a, b

x = 2a   and   y  = 2b

xy = 2a × 2b

= 4ab

= 2 (2ab)

= 2c,   for some integers

So, xy is even

Hence, by contraposition we can  say xy is odd

Therefore, By direct proof and contraposition, the product xy is odd if and only if x and y is odd.

 

 

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