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Proving one set is a subset of another set

Given sets A and B we say that is is a subset of B if every element of A is also an element of B, that is,

x∈A implies x∈B


Basic Outline of the Proof that A is a subset of B:

· Suppose x ∈ A

1. Say what it means for x to be in A

2. Mathematical details

3. Conclude that x satisfies what it means to be in B

· Conclude x∈B


Example

Let A be the set of scalars divisible by 6 and let B be the even numbers. Prove that A is a subset of B.

· Suppose x ∈ A:

1. What it means for x to be in A: x = 6k for any scalar k

2. x = 2 × (3k)

  3k = C

3. What it means for x to be in B: x = 2C

· Conclude x∈B

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