VerifiedHint: In this question, first of all, recollect all the terms like a universal set, subsets, complement of a set, etc. Now, draw the universal set and its subset according to the given information.
Complete step-by-step answer:
Before proceeding with this question, let us consider a few terms.
Universal set: A universal set is a set that contains all the elements or objects of other sets, including its own elements. It is usually denoted by the symbol ‘U’. For example, if we have 2 sets A = {a, b, c, d}, B = {i, j, k} then our universal set is U = {a, b, c, d, i, j, k}.
Subset: A set A is a subset of another set B if all elements of the set A are elements of the set B. We can also say that a set A is contained inside the set B. The subset relationship is denoted as \[A\subset B\]. For example, if we have a set A = {m, n, o, p} then some subsets of set A are {m, n}, {m, o, p} etc.
The complement of the set: Basically in set theory, the complement of a set A refers to elements not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A. For example if we have our universal set as U = {a, b, c, d, i, j, k} and set A = {a, i, c, d} then we get a complement of set A as A’ = {b, j, k}
Now, let us consider our question. First of all, let us draw the Venn diagram for the universal set U = {1, 2, 3, 4, 5, 6, 7}.
Now, let us draw the subsets A = {1, 3, 5, 7} inside the above set.
Now, we know that the complement of A refers to the elements, not in A but is in U. So, these elements are 2, 4, 6. So, A’ = {2, 4, 6}.
In the above Venn diagram, the shaded portion denotes A’.
Note: Students must note that if we have a universal set as U and its subset is A and its complement is A’. So \[A\cap A'\] is null because A and A’ could not have a common element. Also, \[A\cup A'\] is always U because when we add them, they give all the elements that are in the universal set U.
