Question

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

Answer 1

We are given the universal set and the set of which the complement has to be found. We can see which all elements of the given set are absent in the universal set. The set which includes those elements will be the complement of the given set.

Complete step by step solution:
We are given the universal set, $U = \\{ 1,2,3,4,5,6\\}$ and set, $A = \\{ 2,3,4,5\\}$.
We are asked to find $A'$.
$A'$ denotes the complement of a set $A$.
Complement of a set $A$ is defined as the set of all elements in the given universal set $U$ that are not in $A$.
So consider the set $U$.
We can see that,
$1 \in U$. But $1 \notin A$. So by definition of complement, $1 \in A'$.
$2 \in U$. Also $2 \in A$. So by definition of complement, $2 \notin A'$.
$3 \in U$. Also $3 \in A$. So by definition of complement, $3 \notin A'$.
$4 \in U$. Also $4 \in A$. So by definition of complement, $4 \notin A'$.
$5 \in U$. Also $5 \in A$. So by definition of complement, $5 \notin A'$.
$6 \in U$. But $6 \notin A$. So by definition of complement, $6 \in A'$.
So we have, $1,6 \in A'$.

Therefore, $A' = \\{ 1,6\\}$.

Note:
Complement of a set $A$ is defined as the set of all elements in the given universal set $U$ that are not in $A$. So we cannot write the complement of a set if the universal set is not given. But in certain cases, it is understood. For example, if the set considered is the positive real numbers, then the universal set is clearly the set of all real numbers. So the complement of the set contains zero and the negative real numbers.