Question

Let A and B are subsets of U. Identify whether the given statement is right or wrong. (A/B)'= A'/B'

Answer 1

We will first recall the concept of complement, relative complement and universal set from set theory to solve the question. Since, we know that $A'=U-A$ and $A/B=A-B$ , we will use them to solve the above question.

Complete step by step answer:
We can see that the above question is of set theory and from the set theory we know that the universal set is a set which contains all other subsets and is the parent of all sets. Since, from the above question we know A and B are subsets of U. So, U will be the universal set.
Now, complement of any set A, denoted as ${{A}^{c}}\text{or }A'$ is the set which belongs to U but does not belong to A.
And, relative complement is defined with respect to two sets. Let A and B be two sets then, relative complement of set B with respect to set A, is the difference of set A and B and is denoted as A/B, is the set which contains the element belongs to A but does contains element which belong to B i.e.
A/B = A – B.
Now, from the question we know that A and B are two sets which are subsets of U.
So, U is the Universal set of both A and B.
Since, we have to check whether (A/B)'= A'/B' is right or wrong.
So, from LHS we have $\left( A/B \right)'$.
We know that $A/B=A-B$
$\Rightarrow \left( A/B \right)'=\left( A-B \right)'$
And, we know that $A'=U-A$ where U is the Universal set.
Since, A and B are both subsets of U, hence both are contained in U, so we can say (A - B) is also contained in U because (A - B) set contains the element which belongs to A only and not B.
So, we can say that the complement of (A - B) is equal to $U-\left( A-B \right)$.

& \Rightarrow \left( A-B \right)'=U-\left( A-B \right) \\\ & \Rightarrow \left( A-B \right)'=U-A+B \\\ & \Rightarrow \left( A/B \right)'=U-A+B \\\ \end{aligned}$$ Now, we will solve for RHS: From RHS of (A/B)'= A'/B', we have A'/B'. Since, we know that $A/B=A-B$. $\Rightarrow A'/B'=A'-B'$ Now, from the complement we know that the complement of any set A is given as U – A. $\Rightarrow A'=U-A$ and $$B'=U-B$$ Now, we will put the value of A’ and B’ in $A'/B'=A'-B'$. $\Rightarrow A'/B'=A'-B'$ $$\begin{aligned} & \Rightarrow A'/B'=U-A-\left( U-B \right) \\\ & \Rightarrow A'/B'=U-A-U+B \\\ & \Rightarrow A'/B'=U-U-A+B \\\ & \Rightarrow A'/B'=-A+B \\\ & \Rightarrow A'/B'=B-A \\\ \end{aligned}$$ Since, we can see that $\left( A/B \right)'=U-A+B$ and $A'/B'=B-A$ which is not equal. Hence, LHS is not equal to RHS. So, the given statement is wrong. This is our required solution. **Note:** Students are required to note that the relative complement of A with respect to B represented as A/B is basically the ‘A minus B’, means A/B set will contain the element which belongs to A only and not B.