**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.