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

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 proof by exhaustion/cases to prove the following statements.

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^◦.