I used the logic in the title in this answer. Is this implication true and is there a simple explanation of why it is true?

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    $\begingroup$ The question is ill-formed. $\cap$ and $\cup$ are operators that take sets and produce set, but $\Rightarrow$ is a logical connective that needs to be placed between propositions. No matter what kind of things $A$, $B$ and $C$ are, there's no way to get the left-hand side to make sense. $\endgroup$ – Henning Makholm Jul 16 '15 at 14:39
  • $\begingroup$ @HenningMakholm Looking at the original question, it seems it should be usual conjunction/disjunction rather than intersection/union. $\endgroup$ – anakhro Jul 16 '15 at 14:41
  • $\begingroup$ @HenningMakholm Thank you... is this any better? $\endgroup$ – JP McCarthy Jul 16 '15 at 14:42
  • $\begingroup$ ...hmmm. How do I clear up the logic in the answer: math.stackexchange.com/questions/1363311/… $\endgroup$ – JP McCarthy Jul 16 '15 at 14:44
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    $\begingroup$ Note that $\land$ and $\lor$ can be typeset with \land and \lor (or alternatively with \wedge and \vee). $\endgroup$ – Henning Makholm Jul 16 '15 at 14:44

No, this is not a valid implication.

$(A\land B)\Rightarrow (A\lor C)$ is a tautology, and $A\Rightarrow B$ is not. So the former cannot imply the latter.

For example, if $A$ is true but $B$ and $C$ are false, then $(A\land B)\Rightarrow (A\lor C)$ is true, but $A\Rightarrow B$ is false.


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