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.