Question

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

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

Short answer

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

Step-by-step solution

Step: 1

If n=1,

$A_1\cap B=A_1\cap B$

it is true for n=1.

Step: 2

Let P(k) be true.

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

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

Step: 3

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

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