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Question: If the cardinality of set A is \(4\) and that of a set B is \(3\) then what is the cardinality of th...

If the cardinality of set A is 44 and that of a set B is 33 then what is the cardinality of the set AΔBA\Delta B.
A. 11
B. 55
C. 77
D. Cannot be determined

Explanation

Solution

Here we need to know what AΔBA\Delta B means. It refers to all the elements that are there in the set A or the set B but not in their intersection or we can say that all the elements that are there in either set A or set B nut we don’t need to include the elements common to both the set A and B.

Complete Step by Step Solution:
Here we are given that the cardinality of set A is 44 and that of a set B is 33.
We must know the meaning of cardinality. It is actually the number of elements that are there in the set. As here we are given that the cardinality of set A is 44 and that of a set B is 33 this means that there are 44 elements in the set A and there are 33 elements in the set B.
So we can write it as:
n(A)=4 n(B)=3  n\left( A \right) = 4 \\\ n\left( B \right) = 3 \\\
So we need to find the cardinality of AΔBA\Delta B set.
Let us see through Venn diagram that:

We know that AΔBA\Delta B means all the elements that are there in the set A or the set B but not in their intersection or we can say that all the elements that are there in either set A or set B nut we don’t need to include the elements common to both the set A and B.
We can write that:
n(AΔB)=n(A)+n(B)−n(A∩B)n\left( {A\Delta B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)
Here we know the values of n(A) and n(B)n\left( A \right){\text{ and }}n\left( B \right) but not the elements that are common to A and B{\text{A and B}}.
Hence we do not know what n(A∩B)n\left( {A \cap B} \right) is. Therefore the value of n(AΔB)n\left( {A\Delta B} \right) cannot be determined.

Hence we can say that D) is the correct option.

Note:
Here in these types of problems we must know all the symbols that are used in the sets related problems. If we are given the set (B−A)\left( {B - A} \right) then this would mean that all the elements that are there in set B but not in the set A. These problems can be easily solved by using the Venn diagram.