Cartesian product

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Cartesian product and Relation of two sets

 

Cartesian product of two sets

The Cartesian product of the set A with the set B is called the set AxB of all the possible ordered pairs where the first coordinate is from the set A and the second from the set B.

We note: AxB = {(a,b) :a ∈ A\displaystyle \wedge b ∈ B}

Example 1: We have two sets A = {1, 2, 3, 4} and B = {a, b, c}. The Cartesian product of those two sets is:

Cartesian

AxB = \displaystyle \left\{ \left( 1,a \right),\left( 1,b \right),\left( 1,c \right),\left( 2,a \right),\left( 2,b \right),\left( 2,c \right),\left( 3,a \right),(3,b),(3,c),(4,a),(4,b),(4,c) \right\}

Relation

We have seen that every relation of the set A with the set B is a connection that based on a specific rule the elements of the set A are paired with elements of the set B obtaining a subset G of the Cartesian product AxB.

1. The first set A, is called the domain of the relation.

2. The second set B is called the range of the relation.

3. The subset G of the Cartesian product AxB that contains all the ordered pairs (a,b) of the elements that are paired based on a specific rule.

Example 2: We have the relation R with the domain \displaystyle A=\left\{ 6,5,3 \right\} and the range \displaystyle B=\left\{ 2,3,4 \right\} with the rule “ the a is a multiple of b’’.

The Cartesian Product AxB = {(6, 2), (6, 3), (6, 4), (5, 2), (5, 3), (5, 4), (3, 2), (3, 3), (3, 4)}

In this relation, the element 6 ∈ A is paired with the element 2 ∈ B, also with the element 3 ∈ B. The element 5 ∈ A is not paired with any element from the B set. The element 3 ∈ A is paired with the element 3 ∈ B.

The subset G based on the relation we said above is: \displaystyle G=\left\{ (6,2),(6,3),(3,3) \right\}

The demonstration of the subset G in the OXY  plan and with a diagram.

           

Example 3: The relation of the first set A = {1, 2, 3} with the second set B = {2, 4, 6, 7} with the rule that “a is half b“.

a) Write the set G with the given rule.

b)Demonstrate the G set on the Coordinative plan XOY.

c) Demonstrate the set with a diagram.

Solution: Firstly we write the Cartesian product of our two sets:

AxB = {(1, 2), (1, 4), (1, 6), (1, 7), (2, 2), (2, 4), (2, 6), (2, 7), (3, 2), (3, 4), (3, 6), (3, 7)}

a) G = {(1, 2), (2, 4), (3,6)}

b)

c)

 

Example 4: Demonstrate on the plan XOY the graph of the relation y = x² with starting set R and ending set R.

Solution: The Coordinative plan is:

Cartesian product

 

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