SET OPERATIONS
| Site: | Newgate University Minna - Elearning Platform |
| Course: | Elementary Mathematics I |
| Book: | SET OPERATIONS |
| Printed by: | Guest user |
| Date: | Monday, 6 April 2026, 9:27 AM |
Description
In arithmetic, we learn to add, subtract and multiply, that is, we assign to each pair of numbers x and y a number x + y called the sum of x and y, a number x – y called the difference of x and y, and a number xy called the product of x and y.
These assignments are called the operations of addition, subtraction and multiplication of numbers. In this unit, we define the operation Union, Intersection and difference of sets, that is, we will assign new pairs of sets A and B. In a later unit, we will see that these set operations behave in a manner some what similar to the above operations on numbers.
1. Union
The union of sets A and B is the set of all elements which belong to A or to B or to both. We denote the union of A and B by;
A∪B
Which is usually read “A union B”
Example 1.1: In the Venn diagram in fig 2-1, we have shaded A∪B,
i.e. the area of A and the area of B.
Example 1.2: Let S = {a, b. c. d} and T = {f, b, d, g}.
Then S. T = {a, b, c, d, f, g}.
Example 1.3: Let P be the set of positive real numbers and let Q be the set of negative real numbers. The P∪Q, the union of P and Q, consist of all the real numbers except zero. The union of A and B may also be defined concisely by:
A∪B = {xx∈A or x∈B}
Remark 2.1: It follows directly from the definition of the union of two sets that A∪B and B∪A are the same set, i.e.,
A∪B = B∪A
Remark 2.2: Both A and B are always subsets of A and B that is,
A⊂ (A∪B) and B⊂ (A∪B)
In some books, the union of A and B is denoted by A + B and is called the settheoretic sum of A and B or, simply, A plus B
2. Intersection
The Intersection of sets A and B is the set of elements which are common to A and B, that is, those elements which belong to A and which belong to B. We denote the intersection of A and B by:
A∩B
Which is read “A intersection B”.
Example 2.1: In the Venn diagram in fig 2.2, we have shaded A∩B, the area that is common to both A and B
Example 2.2: Let S = {a, b, c, d} and T = {f, b, d, g}. Then S∩T = {b, d}
Example 2.3: Let V = 2, 3, 6, ......} i.e. the multiples of 2; and
Let W = {3, 6, 9,....} i.e. the multiples of 3. Then
V∩W = {6, 12, 18......}
The intersection of A and B may also be defined concisely by
A∩B = {x∈A, x∈B}
Here, the comma has the same meaning as “and”.
Remark 2.3: It follows directly from the definition of the intersection of two
sets that;
A∩B =B∩A
Remark 2.4:
Each of the sets A and B contains A∩B as a subset, i.e.,
(A∩B)⊂A and (A∩B) ⊂B
Remark 2.5: If sets A and B have no elements in common, i.e. if A and B are disjoint, then the intersection of A and B is the null set, i.e. A∩B =∅
In some books, especially on probability, the intersection of A and B is denoted by AB and is called the set-theoretic product of A and B or, simply, A times B.
3. DIFFERENCE
The difference of sets A and B is the set of elements which belong to A but which do not belong to B. We denote the difference of A and B by A – B
Which is read “A difference B” or, simply, “A minus B”.
Example 3.1: In the Venn diagram in Fig 2.3, we have shaded A – B, the area
in A which is n
A – B is shaded
Fig 2.3
Example 3.2: Let R be the set of real numbers and let Q be the set of rational numbers. Then R – Q consists of the irrational numbers.
The difference of A and B may also be defined concisely by
A – B = {xx∈A, x∉B}
Remark 2.6: Set A contains A – B as a subset, i.e.,
(A – B)⊂A
Remark 2.7: The sets (A – B), A∩B and (B – A) are mutually disjoint, that is, the intersection of any two is the null set.
The difference of A and B is sometimes denoted by A/B or A ~ B
4. Complement
The complement of a set A is the set of elements that do not belong to A, that is, the difference of the universal set U and A. We denote the complement of A by A′
Example 4.1: In the Venn diagram in Fig 2.4, we shaded the complement of A, i.e. the area outside A. Here we assume that the universal set U consists of the area in the rectangle.
A’ is shaded
Fig. 2.4
Example 4.2: Let the Universal set U be the English alphabet and let T = {a, b, c}. Then;
T’ = {d, e, f, ....., y, z}
Example 4.3:
Let E = {2, 4, 6, ...}, that is, the even numbers.
Then E′ = {1, 3, 5, ...}, the odd numbers. Here we assume that the universal set is the natural numbers, 1, 2, 3,.....
The complement of A may also be defined concisely by;
A′= {x|x∈U, x∉A} or, simply,
A′= {x|x∉A}
We state some facts about sets, which follow directly from the definition of the complement of a set.
Remark 2.8: The union of any set A and its complement A′ is the universal set, i.e.,
A∪A’ = U
Furthermore, set A and its complement A′ are disjoint, i.e.,
A∩A’ = ∅
Remark 2.9: The complement of the universal set U is the null set ∅, and vice versa, that is,
U’ = ∅ and ∅’ = U
Remark 2.10: The complement of the complement of set A is the set A itself. More briefly,
(A′)′ = A
Our next remark shows how the difference of two sets can be defined in terms of the complement of a set and the intersection of two sets. More specifically,
we have the following basic relationship:
Remark 2.11: The difference of A and B is equal to the intersection of A and the complement of B, that is,
A – B = A∩B′
The proof of Remark 2.11 follows directly from definitions:
A – B = {x|x∈A, x∉B} = {x|x∈A, x∉B’} = A∩B’
5. OPERATIONS ON COMPARABLE SETS
The operations of union, intersection, difference and complement have simple properties when the sets under investigation are comparable. The following theorems can be proved.
Theorem 2.1: Let A be a subset of B. Then the union intersection of A and B is precisely A, that is,
A⊂B implies A∩B = A
Theorem 2.2: Let A be a subset of B. Then the of A and B is precisely B, that is,
A⊂B implies A∪B = B
Theorem 2.3: Let A be a subset of B. Then B’ is a subset of A’, that is,
A⊂B implies B’⊂A’
We illustrate Theorem 2.3 by the Venn diagrams in Fig 2-5 and 2-6. Notice
how the area of B’ is included in the area of A’.
Theorem 2.4: Let A be a subset of B. Then the Union of A and (B – A) is precisely B, that is,
A⊂B implies A∪(B – A) = B
6. SUMMARY
The basic set operations are Union, Intersection, Difference and Complement
defined as:
The Union of sets A and B, denoted by A∪B, is the set of all elements, which belong to A or to B or to both.
The intersection of sets A and B, denoted by A∩B, is the set of
elements, which are common to A and B. If A and B are disjoint then their intersection is the Null set ∅
The difference of sets A and B, denoted by A – B, is the set of elements which belong to A but which do not belong to B.
The complement of a set A, denoted by A’, is the set of elements, which do not belong to A, that is, the difference of the universal set U and A.