Explanation:
(a)
If m is an odd integer, then m + 1 is an even integer.
m (odd integer) m + 1 (even integer)
1 2
3 4
5 6
... ...
Proof:
Suppose m is an odd integer. We can write m as 2n + 1 for some integer n. Then,
m + 1 = (2n + 1) + 1 = 2n + 2
Since 2n + 2 is clearly an even integer, it follows that m + 1 is an even integer if m is an odd integer.
(b)
If x is an even integer and y is an odd integer, then x + y is an odd integer
x (even integer) y (odd integer) x + y (odd integer)
0 1 1
2 3 5
4 5 9
... ... ...
Proof:
Suppose x is an even integer and y is an odd integer. We can write x as 2n and y as 2m + 1 for some integers n and m. Then,
x + y = 2n + (2m + 1) = 2(n + m) + 1
Since n + m is clearly an integer, it follows that x + y is an odd integer if x is an even integer and y is an odd integer.
(c)
If m is an even integer, then 3m^2 + 2m + 3 is an odd integer.
m (even integer) 3m^2 + 2m + 3 (odd integer)
0 3
2 27
4 99
... ...
Proof:
Suppose m is an even integer. We can write m as 2n for some integer n. Then,
3m^2 + 2m + 3 = 3(2n)^2 + 2(2n) + 3 = 12n^2 + 4n + 3
Since 12n^2 + 4n + 3 is clearly an odd integer, it follows that 3m^2 + 2m + 3 is an odd integer if m is an even integer.