Problem 2. Use the proof method specified to prove each statement. Warning: You won’t receive any point if you don’t use the given proof method . Use direct proof to prove the following statements....

Problem 2. Use the proof method specified to prove each statement.

Warning: You won’t receive any point if you don’t use the given proof method.

Use direct proof to prove the following statements.

2(a) The sum of an odd integer and an even integer is odd.

2(b) The product of two odd integers is odd.

Use proof by exhaustion/cases to prove the following statements.

2(c) For every integer
n
such that 0 ≤
n
3, (n
+ 1)²

> n
³.

2(d) For every prime number
x
such that
x
10, 2
<sup>x

+2</sup>
≥ (x
+ 2)².

2(e) For every integer
x,
x
²
+ 3x
+ 1 is odd.

Use contraposition to prove the following statements.

2(f) If
n
is an integer and
n
²
+ 1 is odd, then
n
is even.

2(g) If
x
and
y
are integers and
x
−
y
is odd, then
x
is odd or
y
is odd.

Use contradiction to prove the following statements.

2(h) If there are 15 pigeons in 10 cages, then there is at least one cage containing at least two pigeons.

2(i) If ∠A,∠B,∠C
are interior angles of an acute triangle and ∠A
∠B
∠C, then ∠B >
45^◦.