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**DISCRETE MATH: Prove a "simple" hypothesis involving sets. Use mathematical induction**

## Homework Statement

Prove that if [itex]A_1,\,A_2,\,\dots,\,A_n[/itex] and [itex]B[/itex] are sets, then

[tex]\left(A_1\,\cap\,A_2\,\cap\,\dots\,\cap\,A_n\right)\,\cup\,B\,=\,\left(A_1\,\cup\,B\left)\,\cap\,\left(A_2\,\cup\,B\right)\,\cap\,\dots\,\cap\,\left(A_n\,\cup\,B\right)[/tex]

## Homework Equations

[tex]A\,\cap\,B\,=\,B\,\cap\,A[/tex] <----- commutative law

[tex]A\,\cup\,\left(B\,\cap\,C\right)\,=\,\left(A\,\cup\,B\right)\,\cap\,\left(A\,\cup\,C\right)[/tex] <----- distributive law

## The Attempt at a Solution

I don't know how to start this other than that I need to use the two laws above. Maybe change the notation? I don't know.

[tex]\bigcap_{i\,=\,1}^{n}\,A_i\,\cup\,B\,=\,\left(A_1\,\cap\,B\right)\,\cup\,\left(A_2\,\cap\,B\right)\,\cup\dots\,\cup\,\left(A_n\,\cap\,B\right)[/tex]

What should be the next step or is there a better way of going about this?

NOTE: For LaTeXers, \cup is a union and \cap is an intersection.

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