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Abstract Algebra

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Subset of a Set: Definition and Identification of Subsets

ABSALG-HXYSCK

​Let $A$ and $B$ be two sets.

We say that $A$ is a subset of $B$, written $A\subseteq B$, if $x\in A$ implies $x\in B$.

If $A$ is not a subset of $B$, we write $A\not\subseteq B$.

If $A\subseteq B$ and there are elements of $B$ that are not in $A$, we sometimes write $A\subset B$.

We denote the empty set (the set with no elements) by $\emptyset$.

Two sets $A$, and $B$ are equal, written $A=B$, if and only if they contain the same​ elements. We write $A\not=B$ if they are not equal.

Which of the following is false?

A

$\emptyset\not\subseteq A$ for every set $A$.

B

$\emptyset\subseteq A$ for every set $A$.

C

If $A\subseteq B$ and $B\subseteq A$, then $A=B$.

D

If $A\subseteq B$ and $B\subseteq C$, then $A\subseteq C$.

E

$A\subseteq A$ for every set $A$.