# Counterexamples to the following logical statements about sets

$(1)$ $A \subset B$ or $A \subset C$ $\iff$ $A \subset (B \cup C)$ and

$(2)$ $(A \times B) \subset (C \times D) \implies A \subset C$

Is $(1)$ true? Or does the implication only hold in the forward direction? A friend of mine and I are beginning to work through Munkres's Topology and he is convinced it is only the forward direction, but I think it is a logical equivalence.

Note

Munkres defines "or" to mean "either A or B or both"

Edit 2

$(2)$ was asked after the original question, which was $(1)$. For both questions, Brian M. Scott helped with a counterexample.

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What if $B=\{0\},C=\{1\}$, and $A=\{0,1\}$? –  Brian M. Scott Nov 4 '12 at 20:53
Then it is indeed only a forward implication. Thanks @BrianM.Scott –  Mike Nov 4 '12 at 20:54
You’re welcome! –  Brian M. Scott Nov 4 '12 at 20:55
@BrianM.Scott I can't think of a counterexample to show why $(A \times B) \subset (C \times D) \implies A \subset C$ and $B \subset D$. Wouldn't this implication hold even if $A, B$ were empty? Thanks for all your help by the way. –  Mike Nov 4 '12 at 21:07
Not necessarily, though this one’s a bit tricky. What if $A=\{0\}$, $B=\varnothing$, and $C=D=\{1\}$? –  Brian M. Scott Nov 4 '12 at 21:17

## 1 Answer

A counterexample for (1) is $B=\{0\},C=\{1\}$, and $A=\{0,1\}$.

A counterexample for (2) is $A=\{0\}, B=\varnothing$, and $C=D=\{1\}$.

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+1 ;-) Nice job! –  amWhy Nov 4 '12 at 21:51