Consider the following theorem.
Theorem: For all sets A and B, if A β B, then A βͺ B β B.
Su...
Mathematics, 01.05.2021 18:30 Brucechavez3334
Consider the following theorem.
Theorem: For all sets A and B, if A β B, then A βͺ B β 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 β B. starting point
We must show that A βͺ B β B. conclusion to be shown
Let x be any element in A βͺ 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 β p)
So in either case, x is in B. proof by division into cases
Thus every element in A βͺ B is in B. since x could be any element
of A βͺ B
Therefore, A βͺ B β B [as was to be shown]. ---Select---
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