Consider the following theorem. Theorem: For all sets A and B, if A ⊆ B, then A ∪ B ⊆ B.
Supply explanations for the steps in the proof of the theorem.
Proof:
Statement Explanation
Suppose A, B, and C are any sets such that A ⊆ B. starting point
We must show that A ∪ B ⊆ B. conclusion to be shown
Let x be any element in A ∪ B. start of an element proof
Then x is in A or x is in B. ---Select---
In case x is in A, then x is in B ---Select---
In case x is in B, then x is in B. tautology (p → p)
So in either case, x is in B. proof by division into cases
Thus every element in A ∪ B is in B. since x could be any element
of A ∪ B
Therefore, A ∪ B ⊆ B [as was to be shown]. ---Select---