Assume that n and m are both integers. Prove the following: fn is an odd number

Question

Assume that n and m are both integers. Prove the following: fn is an odd number and m is an even number, then m - n is an odd number. To complete this proof you may use algebra, the logical equivalences, the inference rules, and the following additional "rules": "the definition(s) of even numbers" x is an even number if and only if x can be written as 2k for some integer k x is an even number if and only if n is not an odd number "the definition(s) of odd numbers" x is an odd number if and only if x can be written as 2k 1 for some integer k x is an odd number if and only if n is not an even number Although you are not expected to refer to the algebraic laws that you use by name, you must still SHOW EVERY STEP in detail, even when simply performing algebra.

Explanation / Answer

Give a thumbs up if you like the solution. Let me know in comments, if any clarifications are needed.

The solution starts below this line.

m is an even number. Hence m can be written as

m=2k1

n is an od number. Hence n can be written as

n=2k2+1

Then,

m-n = 2k1 - (2k2+1) = 2k1 - 2k2 - 1

or m-n = 2k1 - 2k2 - 2 + 1

This can be written as

m-n = 2(k1 - k2 - 1) + 1

Here k1, k2 and 1 are integers. Therefore k1 - k2 - 1 will also be an integer. Lets consider this as an integer k

Thus

m-n = 2k + 1

This is the definition of an odd number. Hence m - n is an odd number.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---