Question

Find ${\text{A}}\Delta {\text{B}}$ and by definition: {\text{A}} = \left\\{ {1,2,3,4,5} \right\\} and {\text{B}} = \left\\{ {1,3,5,7} \right\\}.

Answer 1

Here, we need to find the value of the expression ${\text{A}}\Delta {\text{B}}$. First, we will find the elements that come in set ${\text{A}}$ but not in set ${\text{B}}$. Then, we will find the elements that come in set ${\text{B}}$ but not in set ${\text{A}}$. Finally, we will use the formula for ${\text{A}}\Delta {\text{B}}$ to find the elements in the set ${\text{A}}\Delta {\text{B}}$. The $\Delta$ represents the symmetric difference of two or more sets. It includes all those elements that come exactly in one set.

Formula Used:
The expression ${\text{A}}\Delta {\text{B}}$ is given by the union of the elements of the set ${\text{A}} - {\text{B}}$, and the elements of the set ${\text{B}} - {\text{A}}$, that is ${\text{A}}\Delta {\text{B}} = \left( {{\text{A}} - {\text{B}}} \right) \cup \left( {{\text{B}} - {\text{A}}} \right)$.

Complete step-by-step answer:
The expression ${\text{A}}\Delta {\text{B}}$ includes all those elements that come only in set ${\text{A}}$, or only in set ${\text{B}}$. Now, by definition, {\text{A}} = \left\\{ {1,2,3,4,5} \right\\} and {\text{B}} = \left\\{ {1,3,5,7} \right\\}.
The set ${\text{A}} - {\text{B}}$ includes all those elements that come in set ${\text{A}}$ but not in set ${\text{B}}$.
Thus, we get
{\text{A}} - {\text{B}} = \left\\{ {2,4} \right\\}
The set ${\text{B}} - {\text{A}}$ includes all those elements that come in set ${\text{B}}$ but not in set ${\text{A}}$.
Thus, we get
{\text{B}} - {\text{A}} = \left\\{ 7 \right\\}
The expression ${\text{A}}\Delta {\text{B}}$ is given by the union of the elements of the set ${\text{A}} - {\text{B}}$, and the elements of the set ${\text{B}} - {\text{A}}$.
Therefore, we get
${\text{A}}\Delta {\text{B}} = \left( {{\text{A}} - {\text{B}}} \right) \cup \left( {{\text{B}} - {\text{A}}} \right)$
Substituting {\text{A}} - {\text{B}} = \left\\{ {2,4} \right\\} and {\text{B}} - {\text{A}} = \left\\{ 7 \right\\} in the expression, we get
\Rightarrow {\text{A}}\Delta {\text{B}} = \left\\{ {2,4} \right\\} \cup \left\\{ 7 \right\\}
The union of two sets is the set of all the elements in the two sets.
Thus, we get
\Rightarrow {\text{A}}\Delta {\text{B}} = \left\\{ {2,4,7} \right\\}
$\therefore$ We get ${\text{A}}\Delta {\text{B}}$ as \left\\{ {2,4,7} \right\\}.

Note: The expression ${\text{A}}\Delta {\text{B}}$ can be written as the difference in the union and intersection of two sets.
We can also find ${\text{A}}\Delta {\text{B}}$ using the formula ${\text{A}}\Delta {\text{B}} = \left( {{\text{A}} \cup {\text{B}}} \right) - \left( {{\text{A}} \cap {\text{B}}} \right)$.
Since {\text{A}} = \left\\{ {1,2,3,4,5} \right\\} and {\text{B}} = \left\\{ {1,3,5,7} \right\\}, we can see that
\left( {{\text{A}} \cup {\text{B}}} \right) = \left\\{ {1,2,3,4,5} \right\\} \cup \left\\{ {1,3,5,7} \right\\} = \left\\{ {1,2,3,4,5,7} \right\\}
The intersection of two sets is said to be the set of those elements which are common in both the sets.
Therefore, we get
\left( {{\text{A}} \cap {\text{B}}} \right) = \left\\{ {1,2,3,4,5} \right\\} \cap \left\\{ {1,3,5,7} \right\\} = \left\\{ {1,3,5} \right\\}
Now, substituting \left( {{\text{A}} \cup {\text{B}}} \right) = \left\\{ {1,2,3,4,5,7} \right\\} and \left( {{\text{A}} \cap {\text{B}}} \right) = \left\\{ {1,3,5} \right\\} in the formula , we get
\Rightarrow {\text{A}}\Delta {\text{B}} = \left\\{ {1,2,3,4,5,7} \right\\} - \left\\{ {1,3,5} \right\\}
Therefore, we get
\Rightarrow {\text{A}}\Delta {\text{B}} = \left\\{ {2,4,7} \right\\}
$\therefore$ We get ${\text{A}}\Delta {\text{B}}$ as \left\\{ {2,4,7} \right\\}.