Answer to worksheet 5 – Proofs
1) The sum of any two even integers is even.
Proof:
Let a and b be even integers. By definition of even we have that a = 2n and b = 2m.
Consider the sum a + b = 2n + 2m = 2(n + m) = 2k where k = n + m is an integer.
Therefore by definition of even we have shown that my hypothesis is true.
2) The negative of any even integer is even.
Proof:
Let a be an arbitrary even integer and by definition a = 2n for some integer n. Now
consider the -(2n) = 2(-n) = 2k where k = -n is an integer. Therefore by definition -2n
is also an even integer.
3) The sum of any two odd integers is even.
Proof:
Let a and b be odd integers. By definition of odd we have that a = 2n + 1 and
b = 2m + 1. Consider the sum a + b = (2n + 1) + (2m +1) = 2n + 2m +2 = 2k, where
k = n + m + 1 is an integer. Therefore by definition of even we have shown that
a + b is even and my hypothesis is true.
4) If n is odd then n2 is also even.
Counter example: Let n = 3 then (3)2 = 9 is also odd
5) The product of two odd integers is odd.
Proof:
Let n and m be two odd integers. By definition of odd we have that n = 2a + 1
and m = 2b + 1. Consider the product nm = (2a + 1)(2b +1) = 4ab + 2a + 2b +1=
2( 2ab + a + b) + 1 = 2k + 1, where k = (2ab +a +b ) is an integer. Therefore by
definition of odd we have shown that the product of two odd integers is also odd.
6) If n is prime then n2 is prime.
Counter example: Let n = 5 then n2 = 25 which is not prime
7) If 2n + 1 is odd then 2n -1 is also odd
Proof:
Consider 2n – 1 = 2n -2 + 1 = 2(n – 1) + 1 = 2k + 1 where k = n-1 is an integer and
2n -1 is odd by definition.
8) The difference of an even integer minus an odd integer is odd.
Proof:
Let a = 2n be an even integer by definition of even and b = 2k + 1 by definition
of odd and n and k are integers.
Consider a – b = 2n –(2k + 1) = 2(n – k) -1 = 2p-1 where p = n – k is an integer
And where 2p – 1 is odd by proof in problem 7).
9) The sum of two rational numbers is rational:
Proof: Let r and w be two rational numbers. By definition of rational
a
c
r=
and w = where a,b,c,d are integers and b≠0 and d≠0
b
d
a c ad + bc
where bd ≠ 0 is an integer and ad +bc is an
+ =
b d
bd
integer. Therefore the sum is rational by definition.
Consider r + w =
10) The product of two rational numbers is rational.
Proof: Let r and w be two rational numbers. By definition of rational
a
c
r=
and w = where a,b,c,d are integers and b≠0 and d≠0
b
d
a c ac
* =
where bd ≠ 0 is an integer and ac is an
b d bd
integer. Therefore the product is rational by definition.
Consider rw =