# cartesian product of a set | venn diagram math | define cartesian product

In this article **cartesian product of a set** we give the information about **cartesian product of a set,** define cartesian product, **cartesian product meaning**, cartesian product of a sets example.

**Cartesian Product of a Sets:**

Suppose P and Q are two sets such that P is a set of first 3 Even numbers and Q is a set of first 3 Odd numbers,

P = { 2, 4, 6 }

Q = { 1, 3, 5 }

Let’s find the number of pairs of that we can make from these two sets, P and Q. Proceeding in a quite thorough manner, we can recognize that there will be 9 different pairs. They can be written as given below:

(2, 1), ( 2, 3), (2, 5), (4, 1), (4, 3), (4, 5), (6, 1), (6, 3), (6, 5)

The above ordered pairs represent the Cartesian product of given two sets.

#### Cartesian Product Definition:

The Cartesian product of two non-empty sets *P* and *Q* is denoted by P×Q and defined as the “collection of all the ordered pairs (p*,q*) such that p∈P and q∈Q “.

P×Q = { (p,q) : p∈P, q∈Q }

It is also called the cross product, set direct product or the product set of P and *Q* .

**Note:** The collection of ordered pairs it is important matter. By ordered pair, it is meant that two elements taken from each set are written in particular order. So, if p ≠ q , ordered pairs (*p,q*) and *(q,p) *are distinct.

**Cartesian Product of a Sets:**

**Example:**

**Ex 1)** If A = {a, b} and B = {b, c, d, e} then evaluate *AxB and BxA*

**Solution: **

*AxB = { (a,b), (a,c), (a,d), (a,e), (b,b), (b, c), (b, d), (b,e)}*

*BxA = { (b,a), (b,b), (c,a), (c,b), (d,a), (d,b), (e,a), (e,b)}*

* Ex 2) *

*If P = { 1,2,3} and Q = {4, 5} then evaluate*

*PxQ and QxP*

**Solution:**

*PxQ = { (1,4), (1,5), (2,4), (2,5), (3,4), (3,5)}*

*QxP = {(4,1), (4,2), (4,3), (5,1), (5,2), (5,3)}*

* Ex 3) *

*If P = { 1,2,3} and Q = {a, b} then evaluate*

*PxQ and QxP*

**Solution:**

*PxQ = { (1,1), (1,b), (2,a), (2,b), (3,a), (3,b)}*

*QxP = {(a,1), (a,2), (a,3), (b,1), (b,2), (b,3)}*

**Venn diagram math:**

**Venn diagrams:**

John Venn was the first British logist to use closed Diagram to represent sets. Such a representation is called a ‘Venn diagram’.

Venn diagrams are very useful for solving set-based and set-based examples.

Let us understand the use of Venn diagrams from the following example.

**e.g.** A = { 11, 12, 13, 14, 15} Set A is shown by Venn diagram.

**e.g.** B = {x | x is a letter of the word ‘listen’.}

Venn diagram given alongside represents the set B.

#### Venn Diagram Example:

**Ex 1) **Represent the Intersection of two sets by Venn diagram for the following. **(i)** A ={3, 4, 5, 7, 9} B ={1, 4, 5, 8}

**ii)** P = {a, b, c, e, f} Q ={l, m, n , e, b}

**Ex 2) **Represent the Venn diagram for the following.

i) A ∪ B ii) A ∩ B iii) A – B iv) B – A v) U

**Related Links :**

**complement of a set | types of set theory | Roster method | universal set****Operation on Sets | Union of sets | Intersection of sets | Difference of sets****Age problems tricks || word problems ||age problems questions****average meaning || average problems || average definition**