Definition of Set
A well—defined collection of objects, is called a set.
Sets are usually denoted by the capital letters A, B, C, X, Y and Z etc.
The elements of a set are represented by small letters a, b, c, x, y and z etc.
If a is an element of a set A, then we say that a belongs to A. The phrase belongs to denoted by the Greek symbol ∈ (epsilon).
Thus, in notation form, a belongs to set A is written as a∈A and b does not belongs to set A is written as b∉A.
e. g.,
(i) 6∈ℕ, ℕ being the set of natural numbers and 0∉ℕ.
(ii) 36∈ℙ, ℙ being the set of perfect square numbers, so 5∉ℙ.
[Note] Objects, elements and members of a set are synonymous terms
Notation
Each of the objects in the set is called a member of an element of the set. The objects themselves can be almost anything. Books, cities, numbers, animals, flowers, etc.
Elements of a set are usually denoted by lower-case letters. While sets are denoted by capital letters of English larguage.
The symbol ∈ indicates the membership in a set.
If “a is an element of the set A”, then we write a∈A.
The symbol ∈ is read “is a member of” or “is an element of”.
The symbol ∉ is used to indicate that an object is not in the given set.
The symbol ∉ is read “is not a member of” or “is not an element of”.
If x is not an element of the set A then we write x∉A.
📌 Example 1. Let F={1,2,3,4}.
✍ To notate that 2 is element of the set, we’d write 2∊F.
Notation
Commonly, we will use a variable to represent a set, to make it easier to refer to that set
later.
The symbol∊means “is an element of”.
A set that contains no elements, {}, is called the empty set and is notated ∅.
📌 Ex2. Let G={1,2,3,4,5,6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces:
(i) 5…G (ii) 8…G (iii) 0…G (iv) 4…G (v) 2…G (vi) 10…G
🔑 Answers: (i) 5 ∈G (ii) 8 ∉G (iii) 0 ∉G (iv) 4 ∈G (v) 2 ∈G (vi) 10∉G
📌 Ex3. If H={1,3,5,7,9,11,13,15}, then insert the appropriate symbol ∈ or ∉ in each of the following blank spaces.
(i) 1…H (ii) 6…H (iii) 9…H (iv) 14…H
🔑 Answers: (i) 16∈H (ii) 6∉H (til) 9∈H (iv) 14∉H
Sets need not have just numbers as elements. The set B={T,F} consists of two letters, perhaps representing the values “true” and “false.” The set C={a,e,i,o,u} consists of the lowercase vowels in the English alphabet. The set D={(0,0),(1,0),(0,1),(1,1)}“ has as elements the four corner points of a square on the xy coordinate plane. Thus (0,0)∈D, (1,0)∈D etc., but (1,2)∉D (for instance). It is even possible for a set to have other sets as elements. Consider E={1,{2,3},{2,4}}, which has three elements: the number 1, the set {2,3} and the set {2,4}. Thus 1∈E and {2,3}∈E and {2,4}}∈E. But note that 2∉E, 3∉E and 4∉E.
If X is a finite set, its cardinality or size is the number of elements it has, and this number is denoted as |X|. Thus for the sets above, |B|=2, |C|=5, |D|=4 and |E|=3.
