Question

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

Answer 1

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.