(adsbygoogle = window.adsbygoogle || []).push({}); DISCRETE MATH: Prove a "simple" hypothesis involving sets. Use mathematical induction

1. The problem statement, all variables and given/known data

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]

2. Relevant 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

3. 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**