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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 ∈ Ab ∈ 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:

AxB =

**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 and the range 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:

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:

** **