Question

Definition of Even: An integer n ∈ Z is even if there exists an integer q ∈ Z such that n = 2q.

Definition of Odd: An integer n ∈ Z is odd if there exists an integer q ∈ Z such that n = 2q + 1.

Use these definitions to prove only #5:
2. Prove that zero is even.
3. Prove that for every natural number n ∈ N, either n is even or n is odd.
4. Prove that for every natural number n ∈ N, either −n is even or −n is odd. (Don’t use induction!)
5. Apply exercises 2, 3 and 4 to conclude that every integer is either even or odd.

Answer 1

Page oc. Prove that every integer is either Los odd. Lusing (2), (3) 2 (4) ) even It set of integers = 2 -3, -2, -1,0,1,2,3, . ... } using (2) we can say that o is even. Next consider positive integers using (3) we can say that every eositive inegen (or natural no.) ü either even or. odd. → for. are integers lie -...-3,-2,-1) using (4) -n lie negative integers) either odd or even Hence applying all the statement ( , ) and (y) we get every integer ti either even or odd