Question

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
a)$\left( A\cup B \right)'=A'\cap B'$
b)$\left( A\cap B \right)'=A'\cup B'$

Answer 1

Note: Using set theory and given sets, find the values of $A',C',\left( A\cup B \right)'$ and $\left( A\cap B \right)'$. Substitute these values in the expression to verify and prove that LHS = RHS.

Complete step-by-step answer:

We have been given the Universal set, i.e. it contains all the elements from the universal set.
i.e. U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.

Thus we have A = {2, 4, 6, 8} and B = {2, 3, 5, 7}.

We can find $\left( A\cup B \right)$ which is A union B, and it will contain all elements of A and B.

$\left( A\cup B \right)$= {2, 3, 4, 5, 6, 7, 8}.

$\left( A\cap B \right)$, which is A intersection B, it will contain only the common elements of A and B.

$\left( A\cap B \right)$ = {2}.

A’ represents the complement of A, which contains the elements of the universal set that is not in set A.

$\therefore$ A’ = {1, 3, 5, 7, 9}

Similarly, B’ = {1, 4, 6, 8, 9}

Now let us verify the first case.

(i) $\left( A\cup B \right)'=A'\cap B'$

We got $\left( A\cup B \right)$= {2, 3, 4, 5, 6, 7, 8}.

$\therefore$ \left( A\cup B \right)'=\left\\{ 1,9 \right\\}

We got A’ = {1, 3, 5, 7, 9} and B’ = {1, 4, 6, 8, 9}.

\therefore A'\cap B'=\left\\{ 1,9 \right\\}

Thus we got, $\left( A\cup B \right)'=A'\cap B'$.

Thus LHS = RHS.

Now let us take the second case where we need to prove that, $\left( A\cap B \right)'=A'\cup B'$.

We got, A\cap B=\left\\{ 2 \right\\}.

\therefore \left( A\cap B \right)'=\left\\{ 1,3,4,5,6,7,8,9 \right\\}.

We got A’ = {1, 3, 5, 7, 9} and B’ = {1, 4, 6, 8, 9}.

\therefore A'\cup B'=\left\\{ 1,3,4,5,6,7,8,9 \right\\}

Thus we got that, $\left( A\cap B \right)'=A'\cup B'$.

i.e. LHS = RHS.

$\therefore$ We had verified both the cases.

Note: A’ is the complement of A, which can also be denoted as ${{A}^{C}}$. Be careful when you put symbols of union ($\cup$) and intersection ($\cap$). Don’t mix up symbols and you may get wrong answers, while writing the set be careful to mention all the elements.