Question

If A and B are non empty sets such that \[A \supset B\], then
A.\[B' - A' = A - B\]
B.\[B' - A' = B - A\]
C.\[A' - B' = A - B\]
D.\[A' \cap B' = B - A\]
E.\[A' \cup B' = A' - B'\]

Answer 1

Hint: Here we have found the relation between set A and B. Here we will use the Venn diagram to find the relation. We will represent set A by the larger circle and set B by the smaller circle. These two circles will be concentric as the A is a superset of B.

Complete step-by-step answer:
Here it is given that set A is a superset of set B i.e. \[A \supset B\] . We will use a Venn diagram to represent the relation. We will now draw two concentric circles to show that set A is superset of set B.

Now we will observe each option one by one to find the correct answer.
Let’s check the first option, \[B' - A' = A - B\] now.
Here \[B' - A' = U - B - \left( {U - A} \right)\]
After simplifying the terms, we get
\[B' - A' = A - B\]
Thus, the given relation is correct in option A.
Venn diagram for this relation is shown by the shaded area:-

Also we can infer from the verification of option A that option B and option C is wrong.
Now, we will consider option D, \[A' \cap B' = B - A\].
We know that \[A' \cap B'\] is equal to \[\left( {A \cup B} \right)'\] but is not equal to \[B - A\].
Hence, option D is also wrong.
Here in the last option, it is given \[A' \cup B' = A' - B'\].
The relation given is not correct because the correct value is given by\[A' \cup B' = A' + B'\]
Hence, option E is wrong.
Therefore, the correct option is A.

Note: Here, while solving the question we can make a mistake is getting confused between subset and superset. Both superset and subset are different. A set B is said to be subset of set A when set A contains all the elements of set B. Sometimes they are also equal but a set A is said to be a proper superset of set B if set A contains all the element of set B, but set A and set B can’t be equal.