A set is a collection of well-defined elements. Every element in a set is enclosed between the curly braces and separated by a comma. Cardinality means the size of the set. Sets size is nothing but the number of elements present in the given set. Here we will learn more about the Cardinal Properties of Sets. Get the useful formulas and example questions in the following sections.

## Cardinal Properties of Sets – Definition

The various basic properties of sets deal with the union, intersection of two or three sets. We already know that the cardinal number of sets means the number of elements of members or well defined-objects available in the set. Similarly, cardinal properties of sets handle with the sets union, intersection along with the number of sets properties. The various formulas that describe sets cardinal properties are listed here.

**Formulas**

If A and B are two finite sets, then

- n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
- If A ∩ B = ф , then n(A ∪ B) = n(A) + n(B)
- n(A – B) = n(A) – n(A ∩ B)
- n(B – A) = n(B) – n(A ∩ B)

**Also, Read:**

### Cardinal Properties of Sets Examples

**Question 1:**

It was found that out of 70 students, 20 students have passed in mathematics and failed in science and 30 students have passed in mathematics. How many students passed in science and failed in mathematics? How many students passed both subjects?

**Solution:**

Let M = {Number of students who have passed in mathematics}

S = {Number of students who have passed in science}

Number of students who passed science and failed mathematics = Total number of students – Number of students who passed mathematics

= 70 – 30

= 40

Given that,

n(M – S) = 20, n(M) = 30

Then n(M – S) = n(M) – n(M ∩ S)

n(M ∩ S) = n(M) – n(M – S)

= 30 – 20

= 10

Therefore, the number of students who have passed both subjects is 10 and the number of students who have passed in science and failed in mathematics is 40.

**Question 2:**

If P and Q are two finite sets such that n(P) = 25, n(Q) = 11 and n(P ∪ Q) = 32, find n(P ∩ Q).

**Solution:**

Given that,

n(P) = 25, n(Q) = 11 and n(P ∪ Q) = 32

We know that n(P ∪ Q) = n(P) + n(Q) – n(P ∩ Q)

n(P ∩ Q) = n(P) + n(Q) – n(P ∪ Q)

= 25 + 11 – 32

= 36 – 32

= 4

Therefore, n(P ∩ Q) = 4.

**Question 3:**

In a group of 80 people, 30 like jogging, 20 like swimming and 9 like jogging and swimming both.

Find:

(a) how many like jogging only?

(b) how many like swimming only?

(c) how many like at least one of them?

(d) how many like none of them?

**Solution:**

Given that,

Total number of people in the group = 80

Let J be the set of people who like jogging, S be the set of people who like swimming.

Number of people like jogging n(J) = 30

Number of people like swimming n(S) = 20

Number of people who like jogging and swimming n(J ∩ S) = 9

(a)

The number of people who like jogging only means n(J – S)

n(J – S) = n(J) – n(J ∩ S)

= 30 – 9

= 21

So, 21 people like jogging only.

(b)

The number of people who like swimming only means n(S – J)

n(S – J) = n(S) – n(J ∩ S)

= 20 – 9

= 11

So, 11 people like only swimming.

(c)

The number of people who like at least jogging or swimming means n(J ∪ S)

n(J ∪ S) = n(J) + n(S) – n(J ∩ S)

= 20 + 30 – 9

= 50 – 9

= 41

Therefore, 41 people like at least one of them

(d)

The number of people who like none of them = Total number of people in the group – number of people who like at least one of them

= 80 – 41

= 39

Therefore, 39 people like none of them.

### FAQs on Cardinal Properties of Sets

**1. How do you find the cardinal number of a set?**

The number of elements in a set is called the cardinal number of a set. The cardinal number for an empty or null set is always zero. If the elements of a set A are {1, 2, 41, 55, 68, 78, 90, 52}, then the cardinal number for A is 8 and it is represented as n(A) = 8.

**2. What is the purpose of sets?**

The purpose of sets is to collect related objects together. Sets are important everywhere in mathematics because it uses and refers sets in some or other way.

**3. What are the various cardinal properties of sets?**

The different cardinal properties of sets are n(A ∪ B) = n(A) + n(B) – n(A ∩ B), n(A – B) = n(A) – n(A ∩ B) and if A ∩ B = ф , then n(A ∪ B) = n(A) + n(B).

**4. What is a Cardinal Set?**

The number of distinct members of a set is called the cardinal set. The cardinality of a set A is represented as n(A), it counts the number of different elements in a set. We can find the cardinality only for the finite sets.