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

Jason Farmer 2021-08-20 Answered
Determine whether the following set equivalence is true
\(\displaystyle{\left({A}\cup{B}\right)}\ {\left({A}\cap{C}\right)}={B}\cup{\left({A}\ {C}\right)}\)

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Expert Answer

SchulzD
Answered 2021-08-21 Author has 8573 answers

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.

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