# Determine whether the following set equivalence is true (A \cup B) \ (A \cap C) = B

Determine whether the following set equivalence is true
$$\displaystyle{\left({A}\cup{B}\right)}\ {\left({A}\cap{C}\right)}={B}\cup{\left({A}\ {C}\right)}$$

• Questions are typically answered in as fast as 30 minutes

### Plainmath recommends

• Get a detailed answer even on the hardest topics.
• Ask an expert for a step-by-step guidance to learn to do it yourself.

SchulzD

Step 1
Consider the RHS of the given equation.
$$\displaystyle{B}\cup{\left({A}{C}\right)}={B}\cup{\left({A}\cap{C}^{{{c}}}\right)}$$ [Apply the rule $$\displaystyle{A}{B}={A}\cap{B}^{{{c}}}$$]
$$\displaystyle={\left({B}\cup{A}\right)}\cap{\left({B}\cup{C}^{{{c}}}\right)}$$ [Distributive Law]
Consider the LHS of the given equation as follows.
$$\displaystyle{\left({A}\cup{B}\right)}{\left({A}\cap{C}\right)}={\left({A}\cup{B}\right)}\cap{\left({A}\cap{C}\right)}^{{{c}}}$$ [Apply the rule $$\displaystyle{A}{B}={A}\cap{B}^{{{c}}}$$]
$$\displaystyle={\left({A}\cup{B}\right)}\cap{\left({A}^{{{c}}}\cup{C}^{{{c}}}\right)}$$ [Apply De Morgan's Law]
Step 2
Construct truth table for LHS and RHS as follows.
$$\begin{array}{|c|c|} \hline A&B&C&C^{c}&A \cup B&A^{c} \cup C^{c}&(A \cup B) \cap (B \cup C^{c}) \\ \hline T&T&T&F&T&F&F \\ \hline T&T&F&T&T&T&T\\ \hline T&F&T&F&T&F&F\\ \hline F&T&T&F&T&T&T\\ \hline T&F&F&T&T&T&T\\ \hline F&T&F&T&T&T&T\\ \hline F&F&T&F&F&T&F\\ \hline F&F&F&T&F&T&F\\ \hline \end{array}$$
$$\begin{array}{|c|c|} \hline A&B&C&C^{c}&A \cup B&A^{c} \cup C^{c}&(A \cup B) \cap (A^{c} \cup C^{c}) \\ \hline T&T&T&F&T&T&T\\ \hline T&T&F&T&T&T&T\\ \hline T&F&T&F&T&F&F\\ \hline F&T&T&F&T&T&T\\ \hline T&F&F&T&T&T&T\\ \hline F&T&F&T&T&T&T\\ \hline F&F&T&F&F&F&F\\ \hline F&F&F&T&F&T&F\\ \hline \end{array}$$
Observe the last columns of the both truth tables. The first truth value of the first row last column of both truth tables are not the same.
Thus the given set equivalence is not true.