Question

Consider the following relations
1. $A-B=A-\left( A\bigcap B \right)$
2. $A=\left( A\bigcap B \right)\bigcup \left( A-B \right)$
3. $A-\left( B\bigcup C \right)=\left( A-B \right)\bigcup \left( A-C \right)$
Which of these is/are correct?

Answer 1

In this type of question we have to use the concept of set theory. In case of set theory we know that, if $A\And B$ are any two sets then, $A-B=A\bigcap B'$ where $B'$ is the complement of set $B$. Also we know that if $A$ is a set then $A\bigcap A'=\varphi$ and $A\bigcup A'=U$ where $\varphi$ represents empty set and $U$ represents universal set. By De-Morgan’s Law we have, $\left( A\bigcap B \right)'=\left( A'\bigcup B' \right)$ and by distributive law $A\bigcap \left( B\bigcup C \right)=\left( A\bigcap B \right)\bigcup \left( A\bigcap C \right)$. Here we consider each of the statements separately and then we check whether it is correct or not.

Complete step by step answer:
Now we have given
1. $A-B=A-\left( A\bigcap B \right)$
2. $A=\left( A\bigcap B \right)\bigcup \left( A-B \right)$
3. $A-\left( B\bigcup C \right)=\left( A-B \right)\bigcup \left( A-C \right)$
And we have to check which statement is/are correct
Let us consider the each statement one by one and check whether they are correct or not
Let us start with the first statement
1. $A-B=A-\left( A\bigcap B \right)$
$\Rightarrow R.H.S.=A-\left( A\bigcap B \right)$
As we know, $A-B=A\bigcap B'$ where $B'$ is the complement of the set $B$.
$\Rightarrow R.H.S.=A\bigcap \left( A\bigcap B \right)'$
Now by De-Morgan’s theorem we have, $\left( A\bigcap B \right)'=\left( A'\bigcup B' \right)$
$\Rightarrow R.H.S.=A\bigcap \left( A'\bigcup B' \right)$
By using distributive law i.e. $A\bigcap \left( B\bigcup C \right)=\left( A\bigcap B \right)\bigcup \left( A\bigcap C \right)$ we can write,
$\Rightarrow R.H.S.=\left( A\bigcap A' \right)\bigcup \left( A\bigcap B' \right)$
Now as we know that $A\bigcap A'=\varphi$ we get,

& \Rightarrow R.H.S.=\varphi \bigcup \left( A\bigcap B' \right) \\\ & \Rightarrow R.H.S=\left( A\bigcap B' \right) \\\ \end{aligned}$$ Again we will use $$A-B=A\bigcap B'$$ and hence we get, $$\Rightarrow R.H.S.=A-B$$ But we have given $$\Rightarrow L.H.S.=A-B$$ $$\Rightarrow L.H.S.=R.H.S.$$ $$\Rightarrow A-B=A-\left( A\bigcap B \right)$$ Hence, statement 1 is correct.  Now, let us consider the second statement 2\. $$A=\left( A\bigcap B \right)\bigcup \left( A-B \right)$$ To check whether this statement is correct or not let us start with its R.H.S. $$\Rightarrow R.H.S.=\left( A\bigcap B \right)\bigcup \left( A-B \right)$$ Now as we have $$A-B=A\bigcap B'$$ we can rewrite the above expression as $$\Rightarrow R.H.S.=\left( A\bigcap B \right)\bigcup \left( A\bigcap B' \right)$$ Now we will use distributive law i.e. $$A\bigcap \left( B\bigcup C \right)=\left( A\bigcap B \right)\bigcup \left( A\bigcap C \right)$$ $$\Rightarrow R.H.S.=A\bigcap \left( B\bigcup B' \right)$$ But as we know that if $$B$$ is any set then $$\left( B\bigcup B' \right)=U$$ where $$U$$ is the universal set $$\begin{aligned} & \Rightarrow R.H.S.=A\bigcap U \\\ & \Rightarrow R.H.S.=A \\\ \end{aligned}$$ By given $$\Rightarrow L.H.S.=A$$ $$\Rightarrow L.H.S.=R.H.S.$$ $$\Rightarrow A=\left( A\bigcap B \right)\bigcup \left( A-B \right)$$ Hence, the second statement is also correct.  Now, we will check for the third statement 3\. $$A-\left( B\bigcup C \right)=\left( A-B \right)\bigcup \left( A-C \right)$$ In this case we will simplify L.H.S. and R.H.S separately So let us consider L.H.S. $$\begin{aligned} & \Rightarrow L.H.S=A-\left( B\bigcup C \right) \\\ & \Rightarrow L.H.S=A\bigcap \left( B\bigcup C \right)'\cdots \cdots \cdots \left\\{ A-B=A\bigcap B' \right\\} \\\ & \Rightarrow L.H.S=A\bigcap \left( B'\bigcap C' \right)\cdots \cdots \cdots \left( DeMorgan's\text{ }Law \right) \\\ & \Rightarrow L.H.S.=\left( A\bigcap B' \right)\bigcap \left( A\bigcap C' \right)\cdots \cdots \cdots \left( Distributive\text{ }\Pr operty \right) \\\ \end{aligned}$$ Now we will simplify R.H.S. $$\begin{aligned} & \Rightarrow R.H.S.=\left( A-B \right)\bigcup \left( A-C \right) \\\ & \Rightarrow R.H.S=\left( A\bigcap B' \right)\bigcup \left( A\bigcap C' \right)\cdots \cdots \cdots \left\\{ A-B=A\bigcap B' \right\\} \\\ \end{aligned}$$ Here we can easily observe that, $$\Rightarrow L.H.S.\ne R.H.S.$$ Hence, statement 3 is incorrect $$\Rightarrow A-\left( B\bigcup C \right)\ne \left( A-B \right)\bigcup \left( A-C \right)$$ Thus, we can say that statements 1 and 2 are correct statements. **Note:** In this type of question students must be clear with the properties of set theory. Also students have to remember De-Morgan’s Law and Distributive property of set theory. Students have to note that though the third statement is incorrect the corresponding correct statement is $$A-\left( B\bigcup C \right)=\left( A-B \right)\bigcap \left( A-C \right)$$.