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

Hint: 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 \[\left\{ {2,4,7} \right\}\].