Question

Prove that if $A_1,A_2,\dots ,A_n$and B are sets, then

$\begin{array}{r}\left(A_1\cap A_2\cap ,\dots .\cap A_n\right)\cup B\\ =\left(A_1\cup B\right)\cap \left(A_2\cup B\right)\cap \dots \cap \left(A_n\cup B\right)\end{array}$

Short answer

$\begin{array}{r}\left(A_1\cap A_2\cap \dots \dots \cap A_n\right)\cup B\\ =\left(A_1\cup B\right)\cap \left(A_2\cup B\right)\cap \dots \cap \left(A_n\cup B\right)\end{array}$

Step-by-step solution

Step: 1

If n=1,

$A_1\cup B=\left(A_1\cup B\right)$

it is true for n=1.

Step: 2

Let P(k) be true.

$\begin{array}{r}\left(A_1\cap A_2\cap ,\dots .\cap A_k\right)\cup B\\ =\left(A_1\cup B\right)\cap \left(A_2\cup B\right)\cap \dots \cap \left(A_k\cup B\right)\end{array}$

We need to prove that P(k+1) is true.

Step: 3

$\begin{array}{r}\left(A_1\cap A_2\cap ,\dots \cap A_k\cap A_{k+1}\right)\cup B\\ =\left.\left(A_1\cap A_2\cap ,\dots \cap A_k\right)\cap A_{k+1}\right)\cup B\\ =\left(\left(A_1\cap A_2\cap ,\dots \cap A_k\right)\cup B\right)\cap \left(A_{k+1}\cup B\right)\\ =\left(A_1\cup B\right)\cap \left(A_2\cup B\right)\cap \dots \cap \left(A_k\cup B\right)\cap \left(A_{k+1}\cup B\right)\end{array}$

Hence it is true for P(k+1).