Consider the set of integers. 1. Use a direct proof to show that the sum of two even integers is even. 2. Prove that if m + n and n + p are even integers, where m, n, and p are integers, then m + p is even. What kind if proof did you use? 3. Use a direct proof to show that every odd integer is the difference of two squares.
Consider the set of integers.
1. Use a direct proof to show that the sum of two even integers is even.
2. Prove that if m + n and n + p are even integers, where m, n, and p are integers, then m + p
is even. What kind if proof did you use?
3. Use a direct proof to show that every odd integer is the difference of two squares.
4. Prove that if n is an integer and 3n + 2 is even, then n is even using
a) A proof by contraposition.
b) A proof by contradiction.
5. Prove that if n is an integer, the following four statements are equivalent:
a) n is even.
b) n + 1 is odd.
c) 3n + 1 is odd.
d) 3n is even.
6. Find a counterexample to the statement “Every positive integer can be written as the sum
of the squares of three integers”.
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