Examples

Here are some examples of proofs of some basic facts about sets.

First example

Show that \(A ⊆ A ∪ B\).

The definition of subset says that this holds if everything in \(A\) is also in \(A ∪ B\). The definition of union says that any \(x ∈ A ∪ B\) if \(x ∈ A\) or \(x ∈ B\). Thus any \(x ∈ A\) is such that \(x ∈ A ∪ B\), as needed.

Second example

Let \(A, B ⊆ C\), and let’s introduce the notation ‘\(¬X\)’ for \(C - X\).

Show that \(¬(A ∪ B) ⊆ ¬A ∩ ¬B\).

Assume \(x ∈ ¬(A ∪ B)\) (for arbitrary \(x\)); then, our goal is show that \(x\) must be in \(¬A ∩ ¬B\). By definition, \(x ∈ C\), but \(x ∉ A ∪ B\). Hence, \(x ∉ A\) (or else, we would have that \(x ∈ A ∪ B\)). But because, by hypothesis, \(x ∈ C\), we now have that \(x ∈ ¬A\). A similar argument applies to \(¬B\)! Thus \(x ∈ ¬A ∩ ¬B\). Since \(x\) is arbitrary, the argument extends to any element of \(¬(A ∪ B)\), as needed to show that the two sets are in a subset relation.