Exercise 279Logic, sets and proofs › Sets and mapsHigh schoolAdded 3 Nov 2026

Subset relations

#subset#inclusion#definition

Statement

Define $A\subseteq B$ and prove that $\emptyset\subseteq A$ for every set $A$ .

Hint

$A\subseteq B$ means every element of $A$ is an element of $B$ . For the empty set, prove by contradiction.

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Solution

Definition. $A\subseteq B\iff\forall x,\ (x\in A\Rightarrow x\in B)$ .

Empty set. Suppose $\emptyset\not\subseteq A$ . Then there exists $x\in\emptyset$ with $x\notin A$ . But $\emptyset$ has no elements, contradiction.

Conclusion. $\emptyset\subseteq A$ for every set $A$ (vacuous truth).

Subset relations

Statement

Solution

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