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CAT 2025 Lesson : Set Theory - Set Notations

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Most entrance exams (CAT, XAT, IIFT) have questions on Set theory and Venn diagrams. Questions on these could be asked under Quantitative Ability or Data Interpretation sections. This section covers set theory involving mathematical objects and basics of Venn diagrams. The Venn Diagrams lesson under Data Interpretation section would cover advanced concepts.

1. Sets and their Elements

A set is simply a collection of objects, which can be anything. These objects are also called elements or members. Going forward, we will refer to these as elements.

Let us understand the denotations that are commonly used with an example. Let Set A comprise of all the large electronic appliances in your house. If there are three such appliances - Fridge, TV and Washing machine, then the following are true

(a) A == {Fridge, TV, Washing Machine}
The above is a representation of Set A in the roster method, which is detailed in the subsequent section.

(b) Fridge ∈\in A,
This is read as “Fridge is an element of Set A”. This means fridge is an object in Set A.

(c) Chair ∉\notin A,
This is read as “Chair is not an element of Set A”. This means chair is not an object in Set A.

(d) n(A)=3n(A) = 3,
This is the cardinal number or cardinality of Set A, which is the number of distinct elements in Set A.

Example 1

Let's consider a set A =={1,51, 5, computer, 7979, board, 55} and set B = {83,10183, 101} Select the incorrect statement from the options below.

(1) Computer ∈\in A
(2) Watch ∉\notin B
(3) Cardinality of Set A is 66
(4) Cardinality of Set B is 22

Solution

Computer is an element of A and Watch is not an element of B. So, options (1) and (2) are correct.

Although Set A comprises of 66 elements, the number 55 is repeated. So, the Cardinality or distinct elements of Set A is 55. So, option (3) is incorrect.

Cardinality of Set B is 22, which is correct.

Answer: (3) Cardinality of Set A is 66


1.1 Standard Mathematical Notations in Sets

Note that sets are typically denoted by capital letters, like A, B or C, while elements of a set are denoted by small letters. [The set builder notation, covered in the subsequent section, will require a notation for elements.]

RR is the set of all real numbers.
NN is the set of all natural numbers.
ZZ is the set of all integers.
QQ is the set of all rational numbers.
CC is the set of all complex numbers.
PP is the set of all prime numbers.

In your admission tests, questions involving these notations typically describe them as well. However, note that R and N are the two most commonly used notations and you're expected to know these.

1.2 Representation of Sets

The following are the two common ways of representing sets.

Roster Method: The various elements of the set are individually listed in this method.

For example, if Set A comprises of all the natural numbers less than 1111, then the following is the roster method of set representation ⇒ A == {1,2,3,4,5,6,7,8,9,101, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Set Builder Notation: Where the elements of a set follow a certain pattern or contains a certain property, this method can be used to represent the set. This notation contains a variable, a colon or a vertical bar and the property or rule that applies for the variable, within curly brackets.

In the above example of Set A containing the first 1010 natural numbers, the same can be written as A == {x:xx : x ∈\in N,x<11N, x < 11} or A == {x ∣ xx \ | \ x ∈\in N,x<11N, x < 11}

This is to be read as Set A comprises of all values that the variable x can take (which is before the colon or vertical bar), given the properties (which is after the colon or vertical bar). (Indian entrance admission tests typically use colon “ : ” as the separator.)

This set builder notation is very useful in representing sets that involve a lot of elements and in some cases infinite elements, with defined rules/properties. The table below shows how to read a few example sets.

Set Builder Notation Read as
A == {p:p∈R,−3<p<3p : p \in R, -3 < p < 3} Set A contains real numbers that are greater than −3-3 and less than 33.
B == {q:q∈Z,−1000<q<1000q : q \in Z, -1000 < q < 1000} Set B contains integers that are greater than −1000-1000 and less than 10001000.
C == {n2:n∈N,n≤100n^{2} : n \in N, n \le 100} Set C contains squares of the first 100100 natural numbers. Note that the variable(s) before the colon can be in any form of algebraic expression.
D == {x:n∈N,x=2nx : n \in N, x = 2n} Set D contains all even natural numbers. Note that nn is an element of NN and the set E contains all xx, where x=2nx = 2n. So, a set builder could involve more than one variable.
E == {p/q:p,q∈Z,q≠0p/q : p,q \in Z, q \ne 0} Set E contains all rational numbers.


Note that in the tests, there will be questions like X == {1,2,3,...,1001, 2, 3, ..., 100}. This is in roster form. However, you will have to understand the sequence and interpret. In this case, Set X contains the first 100100 natural numbers.

Another example is Y == {1,5,9,13,...,4011, 5, 9, 13, ..., 401}. The numbers have a common difference of 44. These are numbers written as 4k+14k + 1, where kk takes integral values from 00 to 100100.

Example 2

If A == {x3:xx^{3} : x ∈\in Z, x2=4x^{2} = 4}, then A ==

(1) {22}     (2) {2,−22, -2}     (3) {88}   (4) {8,−88, -8}

Solution

Set A comprises of all values of x3x^{3} where x2=4x^{2} = 4

If x2=4x^{2} = 4, then x=+2x = +2 or −2-2.
∴ x3=+8x^{3} = +8 or −8-8
A == {8,−88, -8}

Answer: (4) {8,−88, -8}


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