Question

Suppose $U=\left\{ 1,2,3,4,5,6,7,8,9 \right\},A=\left\{ 1,2,3,4 \right\}$ and $B=\left\{ 2,4,6,8 \right\}$. How ${{\left( A\cup B \right)}^{'}}$ is related to ${{A}^{'}}$ and ${{B}^{'}}$ ? What relation you see between ${{\left( A\cap B \right)}^{'}}$ and ${{A}^{'}}$ and ${{B}^{'}}$ ?

Answer 1

Hint: Find the sets ${{A}^{'}}\cap {{B}^{'}}$, ${{A}^{'}}\cup {{B}^{'}}$, ${{A}^{'}}$, ${{B}^{'}}$, ${{\left( A\cup B \right)}^{'}}$ and ${{\left( A\cap B \right)}^{'}}$. $\left( A\cap B \right)$ means elements common to A and B both and $A\cup B$ means elements common to both as well as not, it means all the elements of A and B will lie in $A\cup B$ (at once). Now, find ${{A}^{'}}\cap {{B}^{'}}$ and ${{\left( A\cup B \right)}^{'}}$ and observe the relation. Similarly, observe the relation between ${{\left( A\cap B \right)}^{'}}$ and ${{A}^{'}}\cup {{B}^{'}}$ .

Complete step-by-step answer:
Given sets from the problem are
$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\}$
$A=\left\{ 1,2,3,4 \right\}$
$B=\left\{ 2,4,6,8 \right\}$
And hence, we need to relate ${{\left( A\cup B \right)}^{'}}$ and ${{\left( A\cap B \right)}^{'}}$ to ${{A}^{'}}$ and ${{B}^{'}}$.
So, let us calculate ${{\left( A\cup B \right)}^{'}},{{\left( A\cap B \right)}^{'}},{{A}^{'}}$ and ${{B}^{'}}$ to get relation among them.
For ${{\left( A\cup B \right)}^{'}}$, we need to calculate $\left( A\cup B \right)$ and transpose of it which means the elements in universal set and does not belong to $A\cup B$.
So, $A\cup B$ is given by writing all the elements of A and B i.e. common and non-common both. So, we get $A\cup B$ as
$A\cup B=\left\{ 1,2,3,4,6,8 \right\}$
And hence ${{\left( A\cup B \right)}^{'}}$ is given as the elements which belong to u not $A\cup B$. So, we get
${{\left( A\cup B \right)}^{'}}=\left\{ 5,7,9 \right\}$ …………………………………(i)

Similarly, the value of ${{\left( A\cap B \right)}^{'}}$ can be given as the elements of u set which does not belong to $A\cap B$.
As we know $A\cap B$ contains the elements which are common to A and B both. So, we get $A\cap B$ as
$A\cap B=\left\{ 2,4 \right\}$
Now, ${{\left( A\cap B \right)}^{'}}$ can be given as
${{\left( A\cap B \right)}^{'}}=\left\{ 1,3,5,6,7,8,9 \right\}$ ………………………………….(ii)

Therefore, to get relations among ${{\left( A\cap B \right)}^{'}}$, ${{\left( A\cup B \right)}^{'}}$ and ${{A}^{'}}$ and ${{B}^{'}}$, we need to calculate${{A}^{'}}$ and ${{B}^{'}}$ as well. So, we get
${{A}^{'}}=\left\{ 5,6,7,8,9 \right\}$
${{B}^{'}}=\left\{ 1,3,5,7,9 \right\}$
Now, let us calculate values of ${{A}^{'}}\cap {{B}^{'}}$ and ${{A}^{'}}\cup {{B}^{'}}$ to get the relations. So, we get
${{A}^{'}}\cap {{B}^{'}}=\left\{ 5,7,9 \right\}$ ………………………………………..(iii)
and
${{A}^{'}}\cup {{B}^{'}}=\left\{ 1,3,5,6,7,8,9 \right\}$ ………………………….(iv)
Now, we can observe that ${{\left( A\cup B \right)}^{'}}$ and ${{A}^{'}}\cap {{B}^{'}}$ have the same elements from the equation (i) and (iii). So, both should be equal. Similarly, ${{A}^{'}}\cup {{B}^{'}}$ and ${{\left( A\cap B \right)}^{'}}$ are equal to each other as well from equation (ii) and (iv). Hence, we get relation of ${{\left( A\cup B \right)}^{'}}$ with ${{A}^{'}}$ and ${{B}^{'}}$ as
${{\left( A\cup B \right)}^{'}}={{A}^{'}}\cap {{B}^{'}}$
And relation of ${{\left( A\cap B \right)}^{'}}$ with ${{A}^{'}}$ and ${{B}^{'}}$ as
${{\left( A\cap B \right)}^{'}}={{A}^{'}}\cup {{B}^{'}}$ .

Note: We need to know the demorgan’s law for getting relationship between ${{\left( A\cup B \right)}^{'}}$ or ${{\left( A\cap B \right)}^{'}}$ and ${{A}^{'}}$ and ${{B}^{'}}$. De-morgan’s law is given as
${{\left( A\cap B \right)}^{'}}={{A}^{'}}\cup {{B}^{'}}$
${{\left( A\cup B \right)}^{'}}={{A}^{'}}\cap {{B}^{'}}$
So, if someone knows the above property, then he/she can directly answer the question without solving a single step of the question.
Don’t miss any elements of ${{A}^{'}}$ or ${{B}^{'}}$ or ${{A}^{'}}\cap {{B}^{'}}$ or ${{A}^{'}}\cup {{B}^{'}}$ to prove the relationship between them. So, be careful while writing the elements of these sets.
We need to know the meaning of symbol’s $\cap $, $\cup $, ‘(on a set) to solve these questions efficiently. So, one should know the significance of these signs as well for solving these kinds of questions.