Set Operations

Set operations are a fundamental topic in mathematics. These operations deal with a fixed collection of items (alphabets, numbers, or even gadgets). It is an essential concept that forms the basis of the relation between two items. 

Let us learn more about the set operations.

Sets and Other Important Terms

A set is a group of objects called elements. We can represent a set as a set builder notation, a roster, and a statement. As mentioned earlier, set operations establish the link between two or more groups of elements.

Venn diagrams help understand set operations better. These logical diagrams show the potential relationship between two finite collections of data.

Disjoint Sets

When two sets do not have any elements in common, they form a disjoint pair.

Example:

A = {a, b, c}

B = {d, e, f}

Here, A and B do not have any elements in common.

Subsets

Let us understand subsets with an example. 

A = {a, b, c}

B = {a, b, c, d, e, f}

Here, all elements in set A are also part of Set B. Therefore, we can say A is a subset of B (A⊂B).

Types of Set Operations

There are four types of set operations in mathematics. They are:

1. Union
2. Intersection
3. Complement
4. Set Difference

Let us delve into them one by one.

1. Union of Sets

Suppose we have two sets A and B. Then, their union is equal to the set that contains all the elements present in set A and set B. We can represent this operation as:

A ∪ B = {x: x ∈ A or x ∈ B}

Where,

x is the elements present in both sets A and B.

Example: 

If set A = {1,2,3,4} and B {6,7}

Then, A ∪ B = {1,2,3,4,6,7}

2. Intersection of Sets

When set A intersects set B, the resulting set will contain items common to both sets. 

n(A∩B) = n(A) + n(B) – n(A∪B) gives the number of elements in A∩B. Here, n(X) is the number of elements in set X. 

To better understand set intersection, consider the following example: 

A = {1, 2, 3}

B = {2, 3, 4}

Therefore,

 A∩B = {2, 3}

3. Complement of Sets

An element of a universal set (U) that is not part of a set A is called its complement. A′ or Ac denotes the complement of set A. 

Consider the following example: 

If,

U = {11, 12, 13, 14, 15}

A = {11, 12, 13}

Then,

A′ or Ac = {14, 15}

4. Set Difference

A set difference indicates the subtraction of elements from a set. We can indicate the set difference between sets A and B as A – B. 

For example:

Suppose

A = {2, 3, 4, 5}

B = {4, 5, 6, 7}

Then 

A – B = {2, 3}

Properties of Set Operations

The properties of set operations are similar to the properties of integers. 

Commutative Law: 

It defines the relationship between two sets.

Consider sets C and D. According to commutative law:

C∪D = D∪C

C∩D = D∩C

Associative Law: 

It defines the relationship between C, D, and E.

(C∪D)∪E= C∪(D∪E)

De-Morgan’s Law: 

Consider two sets, C and D. Then, as per this law:

(C ∪ D)’ = C’ ∩ D’ and (C ∩ D)’ = C’ ∪ D’

The other properties include:

• B ∪ B = B
• B ∩ B = B
• B ∩ ∅ = ∅
• B ∪ ∅ = A
• C ∩ D ⊆ C
• C ⊆ C ∪ D

Conclusion

Set operations are similar to basic numerical operations; the only difference is that we perform them on sets. These operations deal with a finite collection of objects. 

We saw the four main types of set operations in this set of operations notes. We also discussed the properties of set operations.