This answer is restricted. Please login to view the answer of this question.Login Now
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
= 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.
Click here to submit your answer.