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Question: If set A have \(4\) elements and set B has \(7\) elements. What can be the minimum number of element...

If set A have 44 elements and set B has 77 elements. What can be the minimum number of elements inA∪BA \cup B?
A. 77
B. 1414
C. 2121
D. 1111

Explanation

Solution

A∪BA \cup B means union of the set A and the set B that means the elements which are in A or in B or in both A and B. For the minimum elements in A∪BA \cup B, maximum elements must be common in both the sets A and B.

Complete step by step solution:
Here we are given that set A has 44 elements and set B has 77 elements and we need to find what can be the minimum number of elements inA∪BA \cup B. So we need to understand the meaning ofA∪BA \cup B and here we must know that the sign between A and B represents the sign of union which means that A∪BA \cup B means union of the set A and the set B that means the elements which are in A or in B or in both A and B.
Let us understand it by the simple example:
If we have the sets A and B as
⇒A=1,2,4 ⇒B=1,5,3,6 ⇒A∪B=1,2,3,4,5,6  \Rightarrow A = \\{ 1,2,4\\} \\\ \Rightarrow B = \\{ 1,5,3,6\\} \\\ \Rightarrow A \cup B = \\{ 1,2,3,4,5,6\\} \\\
So we get that in theA∪BA \cup B, we have put both the elements which were contained in the sets A and B but as 11 was in both the sets so we counted it only once and not twice.
So for the minimum number of elements in A∪BA \cup B we need to have the maximum common elements as it is understood from the example and also from the formula which says.
n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)
So as we know that we are given that set A has 44 elements and set B has 77 elements and we need to find what can be the minimum number of elements inA∪BA \cup B.
So as out of set A and B we have minimum elements in the set A which is 44 so the case is possible if all those four elements come in the common intersection of the set A and set B which means that
⇒n(A)=4 ⇒n(B)=7 ⇒n(A∩B)=4  \Rightarrow n(A) = 4 \\\ \Rightarrow n(B) = 7 \\\ \Rightarrow n(A \cap B) = 4 \\\
So we can get that n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)

n(A∪B)=4+7−4=7n(A \cup B) = 4 + 7 - 4 = 7

Note:
Similarly for finding the maximum value in the set theory of the number of elements in (A∪B)(A \cup B) we need to know that n(A∩B)=0n(A \cap B) = 0
Hence we will get that n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)
n(A∪B)=4+7−0=11n(A \cup B) = 4 + 7 - 0 = 11