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Question: If \[A=\left\\{ 2,4,6,8,10,12 \right\\}\] and \[B=\left\\{ 3,4,5,6,7,8,10 \right\\}\] then find: (...

If A=\left\\{ 2,4,6,8,10,12 \right\\} and B=\left\\{ 3,4,5,6,7,8,10 \right\\} then find:
(i) A-B
(ii) B-A
(iii) (A−B)∪(B−A)\left( A-B \right)\cup \left( B-A \right)

Explanation

Solution

Hint: We know that the difference between two sets A and B i.e. (A-B) is a set of those elements of set A which are not the elements of set B. Also, the union (∪)\left( \cup \right) of two sets is a set which contains all elements that are present in the two sets which we have considered.

Complete step-by-step answer:
We have been given the sets A=\left\\{ 2,4,6,8,10,12 \right\\} and B=\left\\{ 3,4,5,6,7,8,10 \right\\}.
Now, we know that (A-B) is a set of those elements of set A which are not the elements of set B.
\Rightarrow A-B=\left\\{ 2,4,6,8,10,12 \right\\}-\left\\{ 3,4,5,6,7,8,10 \right\\}
Since the elements 4, 6, 8 and 10 of A are common to the set B. So we will conclude these elements from set A in order to get (A-B).
\Rightarrow A-B=\left\\{ 2,12 \right\\}
Similarly, (B-A) represents the set which has elements of set B that are not present in set A. So we can write,
\Rightarrow B-A=\left\\{ 3,4,5,6,7,8,10 \right\\}-\left\\{ 2,4,6,8,10,12 \right\\}
We can see the elements 4, 6, 8 and 10 are common in both sets. So we will exclude them from set B to get (B-A) as,
\Rightarrow B-A=\left\\{ 3,5,7 \right\\}
Now we know that the union of two sets i.e. A∪BA\cup B means a set which contains all elements of A and B.
\Rightarrow \left( A-B \right)\cup \left( B-A \right)=\left\\{ 2,12 \right\\}\cup \left\\{ 3,5,7 \right\\}=\left\\{ 2,12,3,5,7 \right\\}
Therefore the value of \left( A-B \right)=\left\\{ 2,12 \right\\}, \left( B-A \right)=\left\\{ 3,5,7 \right\\} and \left( A-B \right)\cup \left( B-A \right)=\left\\{ 2,12,3,5,7 \right\\}.

Note: Be careful while finding the difference between the sets as there is a chance that we might include the elements of set B also in (A-B) which will be wrong. Also, remember that a set is a well-defined collection of distinct objects. Also, remember that union of any two sets must not contain the common element in repetition.