Question

If \[U = \left\{ {1,2,3,4,5,6} \right\}\] and \[A = \left\{ {2,3,4,5} \right\}\], then find \[A'\].

Answer 1

Hint:
Here, we need to find the set \[A'\]. We will use the given set and the universal set to find the complement of the set A, and hence find the elements in the set \[A'\]. The complement of a set A is the set of all the elements in the universal set, that are not in the set A. It is denoted by the symbol \[A'\].

Complete step by step solution:
The given set \[U = \left\{ {1,2,3,4,5,6} \right\}\] is the universal set.
The set A is defined as \[A = \left\{ {2,3,4,5} \right\}\].
We will use the given set and the universal set to find the complement of the set A, and hence find the elements in the set \[A'\].
We can explain the complement of a set using the example: if A denotes the set of all even numbers, then \[A'\] denotes the set of all odd numbers.
Therefore, we get
\[A' = U - A\]
Substituting the universal set \[U = \left\{ {1,2,3,4,5,6} \right\}\] and the set \[A = \left\{ {2,3,4,5} \right\}\], we get
\[ \Rightarrow A' = \left\{ {1,2,3,4,5,6} \right\} - \left\{ {2,3,4,5} \right\}\]
The difference of two sets is the set of all elements not common in the two sets.
The elements 2, 3, 4, 5 are common in the universal set and the set A.
Therefore, we get
\[ \Rightarrow A' = \left\{ {1,6} \right\}\]

Thus, we get the set \[A'\] as \[\left\{ {1,6} \right\}\].

Note:
We used the terms ‘set’ and ‘universal set’ in the solution. A set is a collection of objects. It is a well-defined collection of objects. For example: A set of odd numbers is a collection of all odd numbers. The number of elements in a set \[A\] is denoted by \[n\left( A \right)\].
A universal set is the collection of all objects of any related sets. For example: If A is the set of all odd natural numbers, and B is the set of all even numbers, then U is the universal set of all natural numbers.