Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$(1)$ $A \subset B$ 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.


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.

share|cite|improve this question
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 – Moderat 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. – Moderat 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
up vote 4 down vote accepted

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

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

share|cite|improve this answer
+1 ;-) Nice job! – amWhy Nov 4 '12 at 21:51

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.