Question

We are given the sets $\varepsilon =\left\{ 1,2,3,4,5,6,7,8 \right\}$, $A=\left\{ 1,2,3,6 \right\}$, $B=\left\{ 1,2,4,7 \right\}$ and $C=\left\{ 1,3,5,7 \right\}$. List the elements of:
1. $A’=$
2. $B’=$
3. $A\cap B=$
4. $B\cup C=$
5. $A-C=$

Answer 1

Hint: Before solving the above question we will briefly discuss about the set. A set is a group of things that belong together. Set has many different meanings. Like a set of even numbers. In mathematics, a set is a collection of distinct elements or members. The elements that make up a set can be any kind of thing.

Complete step-by-step solution:
Now the set given in the above question is:
$\Rightarrow \varepsilon =\left\{ 1,2,3,4,5,6,7,8 \right\}$
And the subsets of the given set are:
$\Rightarrow A=\left\{ 1,2,3,6 \right\}$, $B=\left\{ 1,2,4,7 \right\}$, $C=\left\{ 1,3,5,7 \right\}$
Now we solve the first $A'=$, it is also called the complement of the subset $A$ .In set theory, the complement of a set $A$ are the elements not in $A$.
Now the complement of the set $A$ is
$\begin{align}
  & \Rightarrow A'=\varepsilon -A \\
 & \Rightarrow A'=\left\{ 1,2,3,4,5,6,7,8 \right\}-\left\{ 1,2,3,6 \right\} \\
 & \Rightarrow A'=\left\{ 4,5,7,8 \right\} \\
\end{align}$
Now the second part is given $B'=$ here we have to find the complement of $B$
$\begin{align}
  & \Rightarrow B'=\varepsilon -B \\
 & \Rightarrow B'=\left\{ 1,2,3,4,5,6,7,8 \right\}-\left\{ 1,2,4,7 \right\} \\
 & \Rightarrow B'=\left\{ 3,5,6,8 \right\} \\
\end{align}$
Now the third part is given $A\cap B=$, here we have to find the intersection of the set $A$ and $B$ , it means we have make a set of common elements from both of the set
$\begin{align}
  & \Rightarrow A\cap B=\left\{ 1,2,3,6 \right\}\cap \left\{ 1,2,4,7 \right\} \\
 & \Rightarrow A\cap B=\left\{ 1,2 \right\} \\
\end{align}$
Now the fourth part is given $B\cup C=$, here we have to find the union of the set $B$ and $C$ , It means we have to make a set of those which are in $A$, in $B$ or in both.
$\begin{align}
  & \Rightarrow B\cup C=\left\{ 1,2,4,7 \right\}\cup \left\{ 1,3,5,7 \right\} \\
 & \Rightarrow B\cup C=\left\{ 1,2,3,4,5,7 \right\} \\
\end{align}$
Now the fifth part of the given question is $A-C=$,
$\begin{align}
  & \Rightarrow A-C=\left\{ 1,2,3,6 \right\}-\left\{ 1,3,5,7 \right\} \\
 & \Rightarrow A-C=\left\{ 2,6 \right\} \\
\end{align}$
Hence we get the answers of all the parts of the question are:
1. $A'=\left\{ 4,5,7,8 \right\}$
2. $B'=\left\{ 3,5,6,8 \right\}$
3. $A\cap B=\left\{ 1,2 \right\}$
4. $B\cup C=\left\{ 1,2,3,4,5,7 \right\}$
5. $A-C=\left\{ 2,6 \right\}$

Note: Sometimes we made mistakes in the intersection of any two sets or union of any two sets, so before solving the questions related to the set theory we should know the basic definition of every operation.