# Proving/Disproving set identity $(A\cap B)\cup C= A\cap (B\cup C)$

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Easiest way is normaly using logic expressions:

$$(A\cap B) \cup C = \{x|\ x\in A,x\in B \lor x\in C \}$$

But in your given example you just have to think about it.

Let's assume $A=\{0\}, B=\{0,1\}, C=\{2\}$ :

$$(A\cap B)\cup C = \{0,2\} \not = A\cap(B\cup C) = \{0\}$$

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### Roy Kesserwani

Attending San Jose State University and currently taking an intro class for java programming. I am currently a beginner with no programming experience but I am passionate to teach myself how to program beginning with java.

Updated on August 01, 2022

For any sets $A$, $B$, $C$; $(A\cap B)\cup C= A\cap (B\cup C)$
It is often helpful when proving set equalities to prove $\subset$ and $\supset$ one at a time. This is done by assuming an element is in the (assumed) smaller set, picking apart the information, and trying to reconstruct the pieces that show it is in the (assumed) larger set. To disprove, you usually look for a counterexample. Helpful?