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   

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