Provide a proof of the following statement: For all integers ? and ?, if ? +...

Provide a proof of the following statement:

For all integers ? and ?, if ? + ? is odd, then ? − ? is odd.

Solutions

Expert Solution

Solution:

To prove: for all integers x and y, if x + y is odd then x - y is odd

Explanation:

Proving the statement given:

=>If x + y is odd then there will be 2 cases-

Case 1: x is odd and y is even

Case 2: x is even and y is odd

=>As sum of even and odd numbers can only by odd. If we add 2 odd numbers then it wil always be even and if we add 2 even numbers then it will be always even hence above 2 cases are only possible.

Proving that x - y is odd:

Case 1: when x is odd and y is even.

=>x - y = odd - even

=>x - y = odd number

=>As subtraction of odd and even number always returns odd number hence x - y is odd for case 1.

Case 2: When x is even and y is odd.

=>As subtraction of even and odd number always returns odd number hence x - y is odd for case 2.

=>Hence we have proved our statement on the basis of above statements given.

I have explained each and every part with the help of statements attached to it.

venereology answered 9 months ago

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