Question

Let \[A = \left\{ {1,2,4,5} \right\},{\text{ }}B = \left\{ {2,3,5,6} \right\}\] and \[C = \left\{ {4,5,6,7} \right\}.\] Verify the following identity.
 $ A - (B \cup C) = (A - B) \cap (A - C). $

Answer 1

Hint: To solve this problem, i.e., to verify the given identity, we will one by one solve the terms of the identity using the sets given to us. Then after getting the values, we will first solve the left-hand side of the expression, then right-hand side of the expression, afterwards we will check, if both sides of the values will match then the identity is verified, if not matched then the identity is not verified.

Complete step-by-step answer:
We have been given that \[A = \left\{ {1,2,4,5} \right\},{\text{ }}B = \left\{ {2,3,5,6} \right\}\] and \[C = \left\{ {4,5,6,7} \right\}.\] We need to verify that $ A - (B \cup C) = (A - B) \cap (A - C). $
So, \[A = \left\{ {1,2,4,5} \right\},{\text{ }}B = \left\{ {2,3,5,6} \right\}\] and \[C = \left\{ {4,5,6,7} \right\}.\]
We know that the union of two sets is a new set that will contain all of the elements that are in at least one of the two sets.
Then using this, we get, $ B \cup C = \{ 2,3,5,6\} \cup \{ 4,5,6,7\} = \{ 2,3,4,5,6,7\} $
Hence, $ A - (B \cup C) = \{ 1,2,4,5\} - \{ 2,3,4,5,6,7\} = \{ 1\} $ \[ \ldots \ldots .eq.\left( 1 \right)\]
And then, $ A - B = \{ 1,2,4,5\} - \{ 2,3,5,6\} = \{ 1,4\} $
Also, $ A - C = \{ 1,2,4,5\} - \{ 4,5,6,7\} = \{ 1,2\} $
We also know that the intersection of two sets is a new set that contains all of the elements that are in both sets.
Then using this, we get, $ (A - B) \cap (A - C) = \{ 1,4\} - \{ 1,2\} = \{ 1\} $ \[ \ldots \ldots .eq.\left( 2 \right)\]
Now, we have been given an identity to verify, which is, $ A - (B \cup C) = (A - B) \cap (A - C). $
So, LHS $ = A - (B \cup C) = \{ 1\} $ [using eq.(1)]
RHS $ = (A - B) \cap (A - C) = \{ 1\} $ [using eq.(2)]
Since, LHS \[ = \] RHS $ = \{ 1\} $
Therefore, $ A - (B \cup C) = (A - B) \cap (A - C) $ is verified.

Note: Most of the time students get confused between union of sets and intersection of sets. So, students should always remember that when we take union of two sets, it means a new set that contains all of the elements that are in at least one of the two sets and when we take the intersection of two sets, it means that a new set that contains all of the elements that are in both sets.