Question

If U = $\left\{a,b,c,d,e,f,g,h,i \right\}$ , A = $\left\{ a,b,c,d \right\}$ , B = $\left\{ b,d,f,h \right\}$ and C = $\left\{ a,d,e,f \right\}$ , then find
A. A’
B. B’
C. C’
D. (B’)’
E. $\left( A\cup B \right)'$
F. $\left( A\cap C \right)'$
G. $\left( B-C \right)'$

Answer 1

Hint: First we will understand the meaning of complement of a set, and with the help of that we will find all the values that have been asked and there are some properties of complement that we will also use.

Complete step by step answer:
Universal set: The set containing all objects or elements and of which all other sets are subsets.
Complement of a set: Complement of a set A, denoted by A c, is the set of all elements that belongs to the universal set but does not belong to set A.
Union: The union (denoted by $\cup $ ) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
Intersection: The intersection of two sets has only the elements common to both sets. If an element is in just one set it is not part of the intersection. The symbol is an upside down $\cap $ .
Here we have U as the universal set and A, B, C as the subsets of U.
Now A’ as per the above definition will be: U – A
Therefore, we have A’ = $\left\{ e,f,g,h,i \right\}$
Similarly we can get the set B’ as: U – B
Therefore, we have B’ = $\left\{ a,c,e,g,i \right\}$
Similarly we can get the set C’ as: U – C
Therefore, we have C’ = $\left\{ b,c,g,h,i \right\}$
Now there is one property of complement of a set,
$\left( A' \right)'=A$
Now we will use this property for solving part (d):
(B’)’ = B = $\left\{ b,d,f,h \right\}$
Now, we can get the union of sets A and B as $A\cup B=\left\{ a,b,c,d,f,h \right\}$
Similarly, we can get $\left( A\cup B \right)'=\left\{ e,g,i \right\}$
Now, we can get the union of sets A and C as $A\cap C=\left\{ a,d \right\}$
Similarly, we can get $\left( A\cap C \right)'=\left\{ b,c,e,f,g,h,i \right\}$
When we subtract two sets the common elements of the set which is being subtract is removed. Therefore, we get
B – C = $\left\{ b,h \right\}$
$\left( B-C \right)'=\left\{ a,c,d,e,f,g,i \right\}$

Note: We have used the definition of terms to understand the meaning of the question and to solve them. Some properties and definitions that we have used must be kept in mind. We must solve this question taking time and going in an orderly manner, else there is a high chance that we might miss some elements or use the wrong formula.