Question

If $A = \left\{ {0,1,2,3} \right\}$ and $B = \left\{ {2,3,4,5,6} \right\}$ , then verify that $A - \left( {A - B} \right) = A \cap B$

Answer 1

Hint:
Consider the left-hand and right-hand sides of the given equation separately. Solve for both of them and then compare the set that is represented by both sides. For $'A - B'$, take all the elements in the set A which are not in set B. And \[A \cap B\] represents the intersection of set A and set B.

Complete step by step solution:
Here in this problem, we need to verify the expression $A - \left( {A - B} \right) = A \cap B$ where A and B are the sets defined as $A = \left\{ {0,1,2,3} \right\}$ and $B = \left\{ {2,3,4,5,6} \right\}$. This is defining the elements contained in both the sets.
For verifying the given expression we need to solve both sides of expression separately and then compare both of them.
For the left-side of the equation, we have: $A - \left( {A - B} \right)$
The symbol $'A - B'$ represents the elements that are in set A but not in set B. Therefore $\left( {A - B} \right)$ means the element in $\left\{ {0,1,2,3} \right\}$ but not in $\left\{ {2,3,4,5,6} \right\}$
 This will give us:
$ \Rightarrow A - \left( {A - B} \right) = \left\{ {0,1,2,3} \right\} - \left( {\left\{ {0,1,2,3} \right\} - \left\{ {2,3,4,5,6} \right\}} \right) = \left\{ {0,1,2,3} \right\} - \left\{ {0,1} \right\}$
Now we need to write elements present in $\left\{ {0,1,2,3} \right\}$ but not in $\left\{ {0,1} \right\}$
\[ \Rightarrow A - \left( {A - B} \right) = \left\{ {0,1,2,3} \right\} - \left\{ {0,1} \right\} = \left\{ {2,3} \right\}\]
Therefore, the LHS of the given equation is \[\left\{ {2,3} \right\}\]
For the right-hand side of the given expression, we get:
\[ \Rightarrow A \cap B = \left\{ {0,1,2,3} \right\} \cap \left\{ {2,3,4,5,6} \right\}\]
The symbol $' \cap '$ represents the intersection of two sets. The set operation intersection takes only the elements that are in both sets. The intersection contains the elements that the two sets have in common. The intersection is where the two sets overlap. The intersection of two sets A and B, denoted by \[A \cap B\] is the set containing all elements of A that also belong to B (or equivalently, all elements of B that also belong to A).
\[ \Rightarrow A \cap B = \left\{ {0,1,2,3} \right\} \cap \left\{ {2,3,4,5,6} \right\} = \left\{ {2,3} \right\}\]
Therefore, we get RHS as \[\left\{ {2,3} \right\}\]
Thus, we get $LHS = RHS$

Hence, the set represented by both the sides of the given equation is the same, and therefore, we can say this verifies the given expression $A - \left( {A - B} \right) = A \cap B$

Note:
In questions from set theory it is important to understand the definition of different symbols and operations. An alternative method can be taken by the use of the Venn diagram to solve for both the sides of the given expression. Remember that verification does not mean proofing the equality in the equation, but to put the given value and show that it satisfies the expression.