Sets: It is a collection of distinct objects, also called elements or members. These objects can be anything, such as numbers, letters, or even other sets. Sets are usually denoted by capital letters, and their elements are enclosed within curly braces.
Examples:
1) A = {1, 2, 3, 4, 5}
(Set A consists of the numbers 1, 2, 3, 4, and 5)
2) B = {a, b, c, d}
(Set B consists of the letters a, b, c, and d)
Terminology of Sets:
1) Singleton Set
2) Subsets
3) Proper Subsets
4) Equal Sets
5) Universal Set
6) Finite Set
7) Infinite Set
Singleton Set : It is a set that contains only one element.
Example: A= {5}
Subsets: Subsets are sets that are contained within another set i.e. a part of a set. In other words, all the elements of a subset are also elements of the larger set. If a set A is a subset of a set B, it is denoted as A ⊆ B. If set A is not a subset of B, it is denoted as A ⊈ B.
Note: The empty set, denoted by ∅ or {}, is considered a subset of every set.
Examples:
1) Let A = {1, 2, 3} and B = {1, 2, 3, 4, 5}. In this case, A is a subset of B because all the elements of set A (1, 2, and 3) are also present in set B. So we can write A ⊆ B.
2) Let C = {a, b} and D = {a, b, c, d}. In this case, C is also a subset of D because all the elements of set C (a and b) are present in set D. Thus, C ⊆ D.
3) Let E = {x, 2} and F = {1, 2, 3}. In this case, set E is not a subset of F because one element of set E (i.e. x) are not present in set F. Therefore, E ⊈ F.
4) Let G = {1, 2, 3} and H = {1, 2, 3}. In this case, set G is a subset of H because all the elements of set G are also present in H. So we can write G ⊆ H
Proper subset: It is a subset that contains some, but not all, of the elements of another set. In other words, if every element of set A is also an element of set B, but there exists at least one element in set B that is not in set A, then A is a proper subset of B. It is denoted as A ⊂ B.
Examples:
1) Let A = {1, 2, 3} and B = {1, 2, 3, 4, 5}.
In this case, A is a proper subset of B because all the elements of set A (1, 2, and 3) are also present in set B and Set B also has some additional elements i.e. 4 & 5 . So we can write A ⊂ B.
2) Let C = {a, b} and D = {a, b, c, d}. In this case, C is also a proper subset of D because all the elements of set C (a and b) are present in set D and Set D also has some additional elements i.e. c & d . Thus, C ⊂ D.
3) Let G = {1, 2, 3} and H = {1, 2, 3}. In this case, set G is a subset of H, but not proper subset because all the elements of set G are also present in H, but set H don’t have any additional element. So we can write G ⊆ H , but not G ⊂ H. Here G and H are equal sets and denoted by G=H
Equal sets are sets that have the exact same elements. In other words, two sets are equal if and only if they contain the same elements, regardless of their order or repetition. Equal sets are denoted by the symbol “=”. If all the elements in one set are present in the other set, and vice versa, the sets are considered equal.
Examples:
1) Let A = {1, 2, 3} and B = {3, 2, 1}.
In this case, set A and set B have the same elements, even though the order is different. Therefore, A = B.
2) Let C = {a, b, c} and D = {c, b, a}.
In this case, set C and set D have the same elements, just like in the previous example. Thus, C = D.
3) Let E = {1, 2, 3} and F = {1, 1, 2, 3}.
In this case, even though set F contains repeated elements (two occurrences of 1), it still has the same elements as set E. Therefore, E = F
4) Let G = {1, 2, 3} and H = {1, 2, 3, 4}.
In this case, set G and set H do not have the same elements because set H has an additional element, 4. Thus, G ≠ H.
Universal set: It is a set that contains all the objects or elements under consideration for a particular discussion or problem. It represents the largest possible collection of elements within a given context. The universal set is often denoted by the symbol “U.”
Examples:
1) A = {1, 2, 3, 4, 5}
(Set A consists of the numbers 1, 2, 3, 4, and 5)
then universal sets can be any of the following:
U={set of all natural numbers}
u={Set of all rational numbers}
U={Set of all real numbers}
2) B = {amit, sumit, sunil}
(Set B consists of 3 persons amit, sumit and sunil
then universal set can be any 1 of the following
U ={set of all persons in the class}
U={set of all persons in the school}
U={set of all persons in the city}
Finite sets: are sets that have a countable number of elements. In other words, a set is finite if it contains a specific, definite number of elements. This number can be zero (an empty set) or any positive integer.
Examples:
A = {1, 2, 3, 4, 5}: This set consists of five elements, namely the numbers 1, 2, 3, 4, and 5.
B = {a, b, c}: This set contains three elements, the letters ‘a’, ‘b’, and ‘c’.
C = {dog, cat, rabbit, hamster}: This set includes four elements, representing different types of pets.
D = {apple, banana, orange, mango}: This set has four elements, representing different fruits.
E = {Monday, Tuesday, Wednesday, Thursday, Friday}: This set consists of the names of five weekdays.
All above sets have a finite number of elements, and we can count or list all the elements within the set.
Infinite Sets They have an uncountable number of elements. They continue indefinitely without end.
Examples:
1) Set of Natural Numbers (N):
N = {1, 2, 3, 4, 5, …}
The set of natural numbers includes all positive integers starting from 1 and continuing infinitely.
2) Set of Whole Numbers (W):
W = {0, 1, 2, 3, 4, 5, …}
The set of whole numbers includes all non-negative integers, starting from 0 and continuing infinitely.
3) Set of Integers (Z):
Z = {…, -3, -2, -1, 0, 1, 2, 3, …}
The set of integers includes all positive and negative integers, as well as zero, and it continues infinitely in both directions.
4) Set of Rational Numbers (Q):
Q = {a/b | a, b ∈ Z and b ≠ 0}
The set of rational numbers includes all numbers that can be expressed as a fraction of two integers, where the denominator is not zero. It includes numbers like 1/2, -3/4, 5/1, and so on. The set of rational numbers is infinite.
5) Set of Real Numbers (R):
R includes all rational and irrational numbers. The set of real numbers is an uncountably infinite set that includes all rational numbers, irrational numbers (such as √2 and π), and all possible decimal representations.