For any two sets A and B prove that...
A is a subset of A union B.
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∵ A∩B is set of all that elements that belongs to both A and B. Hence, all elements of A∩B contained in A.
Hence,
A∩B⊂A
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Answer Keys.
This is an identity. The explanation is as follows.
Solution.
Using the Benn diagram,
- is the grey region.
- is the yellow region.
Hence, belongs to the region of . In other words, .
More information.
Proof by Contrapositive
Sets are one of the ways that are used in proofs. Let's take an example.
How do we prove, that for x, y such that the values are ?
At the first glance, this looks complicated, but it is easier to solve with sets.
,
Let the truth sets of two statements be and . We need . If the statement is true, will be true according to the Benn diagram.
We shall test . Now given statement becomes "for the values of x, y such that we have ."
It is true, and this way of proof is proof by contrapositive.
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