# Questions tagged [elementary-set-theory]

This tag is for elementary questions on set theory, spanning topics usually found in introductory courses in set theory, in addition to review sections of graduate textbooks in the same field. Topics include intersections and unions, differences and complements, De Morgan's laws, Venn diagrams, relations and functions, countability and uncountability, etc. More advanced topics should use (set-theory) instead.

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### Is A a subset or and element of set C?

Let A, B, C be 3 sets. If A belongs to B, and B is a subset of C, is it true that A is a subset of C? They say A is a set. So A should be a subset of C. But my textbook says it’s not because A is an ...
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### A doubt on the question $C = f^{-1}(f(C)) \iff f$ is injective and the similar surjective version

Prove that $C = f^{-1}(f(C)) \iff f$ is injective and $f(f^{-1}(D)) = D \iff f$ is surjective I have a doubt in the question asked above. In this statement, $C = f^{-1}(f(C)) \iff f$ is injective I ...
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### How can a counterexample be used to disprove $(A\setminus B)\setminus C \equiv A\setminus(B\setminus C)$? [closed]

How can a counterexample or logical proof be used to prove the following non-equivalence? (established using a Venn diagram) $$(A\setminus B)\setminus C \neq A\setminus(B\setminus C)$$ This arose ...
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### How does the identity function $id_X$ imply injectivity of the right inverse function of the composition $g \circ f$?

$f: X \rightarrow Y$, $g: Y \rightarrow X$, $g \circ f = id_X$ I can see, graphically, how having $f(x) = f(x')$ with $x \neq x'$ would mean that $f(x)$ (or $f(x')$ for that matter) would have to map ...
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### Prove that A-(B⋃C) = (A-B)⋂(A-C)

Please let me know if this proof is right, I think it is but I still want confirmation ...
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### What is the meaning of $\{1,\dots,k \}^n$? [closed]

I'm trying to figure out what the set $\{1,\dots,k \}^n$ is. I know that the number of $n$-tuples formed from the numbers $1,\dots,k$ is the cardinality of that set, but I'm not sure what the ...
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