Intersection

We can say that a Set is the collection of different objects grouped together. Any set is the most basic element used in mathematics. In everyday life we observe that Sets are used to make groups of same type of elements. When we study sets theory, we need to re- organize the group of the sets which always helps us to get the solutions. One of the main applications of the sets theory is to make the Relations between two sets and then observe the elements of the new set. For making these relations, certain operations can be performed on the Numbers. We will discuss these operations one by one.
Intersection of two sets:  We can also form a new set by determining the common elements of the given two sets. This is called intersection of the two sets.  If A∩ B = ∅, then ‘A’ and ‘B’ are said as disjoint setsIntersection of a set is a basic operation performed in the sets Algebra. Here we must first know that as Union represents “ OR ” operation, similarly  Intersection represents “AND” operation, which means the element exist in both the sets,  will belong to the intersection sets.  It means that in order to form the new set with the  intersection sets ( A and B ), the new set will have to coincide in both or more sets.
Here are some of the examples to show the intersection of sets. Let A= 1, 2 and B= red, black then ‘A’ intersection B = φ, Null set. We observe that there is no element common in both the sets, so the result is the null set (a set with no element).
Similarly if A= 1, 2, 3, 4 and B = 4, 5, 6, 7.
Here both sets ‘A’ and ‘B’ have 4 as common, so ‘A’ intersection ‘B’ will be = 4
Again we observe that if set ‘A’ and set ‘B’ have all elements same, then ‘A’ intersection B = A.

Note:

Intersection of two sets is the common term in the given set, if we have two set ‘A’ and ‘B’ and ‘A’ has elements as 1,2,3,4 and set ‘B’ has 3,4,5,6 so intersection will be 3,4, intersection is denoted by the symbol ‘∪’, and if two sets have nothing in common than the set is called as Nullary intersection.

Note:

Arbitrary Intersections of two sets ‘X’ and ‘Y’ refers to the set that contains all the elements of ‘X’ and also belong to ‘Y’ but not the other elements. It is abbreviated as X ∩ Y. If X ∩ Y belongs to ‘R’ then it means that ‘X’ belongs to ‘R’ and ‘Y’ belongs to ‘R’.

What do you mean by Intersection of the two sets? Explain with an example.

Intersection of Sets, represented by A ∩ B. Word Intersection means the elements of the two Sets which are common. So if we write Let A = 1, 2, 3, 4 and B = 4, 5, 6 then we can say that
A ∩ B = 4 which includes all the common elements of Set A and set B. Here we observe that only element 4 is common in both the sets A and B.

Is A ∩ φ = A True or False? Give reason in support of your answer.

A ∩ φ = A is not a True statement. As when we write A ∩ φ, it means the intersecting elements of Set ‘A’ and a null set φ.  Let if A = 1, 3, 5 and we know that null set has no element in it. Then the Intersection of two such Sets is always a null set, because intersection means the common elements of the two Sets. So we come to a conclusion that A ∩ φ = φ.