intersection

In the world of math, the intersection of sets A and B, denoted through A B, is the set consisting of all values of ‘A’ that also incline with B or, equivalently, all values of ‘B’ that also incline to A. 

Provided with two sets, A and B, the intersection is used to find the elements in commonality. It is regularly used in the fields of geometry, set theory, number theory etc.

The intersection between two sets is implicated by the sign “” which is an infix notation.

Let us take two sets and understand through intersection study material format.

A = {1,2,3} and B = {2,3,8}

then

A B = {2,3}

Symbolically the intersection between two sets is defined as below 

Provided with two sets A and B then

A B = {x : x A and x B}

What this implies is that x is a constituent of the intersection A B if and only when x is a constituent of A as well as a constituent of B. 

Intersecting and Disjoint Sets

The study material notes on intersection emphasise on different sets as well. These are interacting and disjoint sets. 

When we have two sets A and B then we can say A intersects (meets) B if and only if there exists some x that is an element of A as well as an element of B. In this case, it can also be established that A intersects (meets) B at x. 

The same can also be rephrased as A intersects B if and only if the set AB is an inhabited set. According to classical mathematics, a set is said to be inhabited if it is non-empty. Therefore, AB is an inhabited set if there exists some x such that x A B.

Set A and B are implied to be disjoint only if A does not intersect B. This implicates that A and B have no constituents of commonality. Symbolically, this can be represented as 

A and B are disjoint if and only if A B = . Here, represents the null set. Let us look at the examples of both intersecting and disjoint sets.

A = {2,3,6,8} and B = {6,8,9,10}

then, A B ={6,8} and it can be said that A intersects B since AB. 

Now let us consider C={1,2,3} and D={4,5,6} here we can say that,

CD=. Since the set CD is an empty or null set, we can say that C and D are disjoint from each other.

Algebraic Properties Of Intersection

In the intersection study material, Intersection follows the associative property. Therefore, between three sets A,B and C , 

A(BC) =(AB)C

Example: 

For three sets A={1,2,3}, B={2,3,4} and C={3,4,5}.

To find

A(BC)  

Now, BC={3,4} 

Thus, A{3,4}=3

Now let us find out

(AB)C

Now, AB={2,3}

Thus, {2,3}C=3

Hence,

A(BC) =(AB)C

Intersection also follows the Commutative property, which means that
AB=BA

The property is self-explanatory since both sets fundamentally remain the same even when their places are interchanged.

Also, for the set, A and , which is the empty set, the intersection is given us

A=

For the set, A and the universal set U, the intersection between the two sets is given as;

AU=A

This holds true because the elements that are part of A are also a part of U, since U is the universal set and contains all the elements of the sample space.

And for any set, intersection is also idempotent, which means that for any set A, 

AA=A

Intersection also distributes over union and that holds true vice versa too, that is, union distributes over intersection.

Therefore 

A(BC)=(AB)(AC)

For a set A, in a universal set U, where the complement of the set A is Ac, then 

AAc=

This is because both the sets are mutually exclusive to each other.

Arbitrary Intersections In Intersection Study Material

The maximum popular notion is the intersection of an arbitrary non-empty series of sets. If M is considered to be set, that is nonempty and whose elements are therefore sets, then x is constituent of the M intersection, only when for each value of A in M, x is a constituent of A. In symbols, it can be expressed as;

(xAMA) (AM,xA)

In simple language in intersection study material, this means that an element x belongs to M if and only if x also belongs to A and A is an element of M for all A. The elements of M are themselves individual sets, and each set is a non-empty set. 

Conclusion

The study material notes on intersection of two sets is denoted by the symbol . The intersection of two sets A and B is built with all the commonality of constituents of the two sets. A is said to intersect B if and only if the set created by the intersection of both the sets is an inhabited set, that is, the set created by their intersection is not an empty set. If an element x belongs to the intersection between two sets A and B then it can be said that the set A intersects the set B at x. 

The intersection study material emphasises how intersection is used in the fields of set theory. The study material notes on intersection lay fundamentals and application of intersection in depth.