Question

Let A = { 1, 2, 4, 5 }, B = { 2, 3, 5, 6 }, C = { 4, 5, 6, 7 }. Verify the following identities:
$A\cap \left( B-C \right)=\left( A\cap B \right)-\left( A\cap C \right)$

Answer 1

Hint: First we will write the definitions of required terms like intersection and union of sets and then we will use them to verify $A\cap \left( B-C \right)=\left( A\cap B \right)-\left( A\cap C \right)$ by substituting the values of A, B and C and show that LHS = RHS.

Complete step-by-step answer:
Union: The union (denoted by $\cup $ ) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
Intersection: The intersection of two sets has only the elements common to both sets. If an element is in just one set it is not part of the intersection. The symbol is an upside down $\cap $ .
When we subtract two sets the common elements of the set which is being subtract is removed and the remaining elements is the final answer.
Let’s start by solving from LHS,
Let’s first find the term B – C ,
We have B = { 2, 3, 5, 6 }, C = { 4, 5, 6, 7 }
Now we have to remove the common elements present in B and C from B.
5 and 6 are the common element, so after removing it we get,
B – C = { 2, 3 }
Now we will find the value of $A\cap \left( B-C \right)$ ,
We have A = { 1, 2, 4, 5 } and B – C = { 2, 3 }, now the common element in both of them is 2.
Therefore,
$A\cap \left( B-C \right)$ = { 2 }
Hence, we have found the value of LHS.
Now we will solve RHS,
Let’s find the value of $A\cap B$,
A = { 1, 2, 4, 5 }, B = { 2, 3, 5, 6 }, the common elements in these two are 2 and 5.
Hence, we get
$A\cap B$ = { 2, 5 }
Now let’s find the value of $A\cap C$
A = { 1, 2, 4, 5 }, C = { 4, 5, 6, 7 } now the common elements in these two are 4 and 5.
Hence, we get
$A\cap C$ = { 4, 5 }
Now we will find the value $\left( A\cap B \right)-\left( A\cap C \right)$
We have $A\cap B$ = { 2, 5 } and $A\cap C$ = { 4, 5 }, now we have to remove the common elements present in both of them from $A\cap B$, which is 5.
After removing we get,
$\left( A\cap B \right)-\left( A\cap C \right)$ = { 2 }
Hence, we have found the value of RHS.
Hence we can see that LHS = RHS.
Hence, $A\cap \left( B-C \right)=\left( A\cap B \right)-\left( A\cap C \right)$ has been verified.

Note: The definitions of union and intersection of sets must be kept in mind while solving the question. One point that should be kept in mind is when we subtract two sets the common elements of the set which is being subtracted is removed and the remaining elements is the final answer. We should not get confused with the union and intersection of sets in these types of questions.