Proof by contradiction(sets)

Mathematics Asked by JASDEEP SINGH on December 12, 2020

I am stuck with this question.

How to prove that compliments of a universal set is the null set by contradiction?

One Answer

Suppose the complement of the universal set is not empty.

Then the complement of the universal set has at least one element. Let $x$ be an element of the the complement.

But $x$ exists. so it is in the universal set.

So $x$ can not be in the complement of the universal set.

==== oops, old answer: I didn't realize you specifically wanted a prove by contradiction=====

It should follow directly by definitions.

If $U$ is the universal set, then $A^c = {xin U| xnot in A}={x| x in U,$ and $x in A}$.

So by definition $U^c = {xin U| xnot in U}={x| xin U, x not in U} = emptyset$.

Answered by fleablood on December 12, 2020

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