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then-ponder · 8 months

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Proposition: suppose x,y are integers. If x is even then xy is even

Symbolic form: ∀x,y ∈ ℤ, P(x) ⟹ Q(x,y)

Proof.

Suppose x and y are integers and x is even

Then y is either even or odd. Let’s consider the cases separately

Case One: y is even.

Suppose y is even. Then y = 2a, a ∈ ℤ.

Thus xy = 2b(2a), b is an integer 2b = x.

So xy = 4ab

Thus xy = 2c, c is the integer c = 2ab

This xy is even by definition of an even number [Def. N2]

Case Two: y is odd

Suppose that y is odd. Then y = 2a + 1, a ∈ ℤ. [Def. N1]

Thus xy = 2b(2a + 1), b is the integer x = 2b

Then, xy = 4ab + 2a

So xy = 2(2ab + a)

Then xy = 2c, c is an integer c = 2ab + a

Therefore xy is even by definition of an even number[ Def. N2].

Thus xy is always even when x is even because in all cases, when x is even then xy is even.

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