Lesson 1: Relation

Video Lesson

Competencies (MLC):

At the end of this lesson, you will be able to:

• Define patterns. arithmetic and geometric pattern.
• Produce sequences of patterns.
• Define a relation as a subset of a Cartesian product of two sets.
• Represent relations using sets of ordered pairs, arrow diagrams, and matrices.
• Determine the domain, range, and codomain of a relation.

Key Terms

Brainstorming Question

1. Write the numbers which come next in 1, 4, 9, 16, ________.

2. Create a table of values for the set of values given in the first row of Table by evaluating the algebraic expression 2n + 3

3. What relationship can be represented by the table given below?

1.1. Revision on Pattern

Definition:

Patterns are defined as regular, repeated, recurring forms or designs identifying relationships, finding logic to form generalizations and make predictions.

Example 1 

a. Square Number Pattern 1, 4, 9, 16, 25,

b. Even natural number pattern 2, 4, 6, 8, 10,…

c. Triangle pattern consists of dots are given below.

Arithmetic pattern

• The arithmetic pattern is also known as the algebraic pattern.
• In an arithmetic pattern, terms are found by adding or subtracting.

Example 2

1. Consider the pattern 4, 11, 18, 25, 32, … It is an arithmetic pattern found by adding 5 for each terms and an increasing sequence.
2. An arithmetic decreasing pattern is given below with constant difference -3.

Geometric pattern

1. The geometric pattern is the sequence of numbers that are obtained by multiplication and division operation.
2. If two or more numbers in the sequence are provided, the next term in the pattern is obtained using multiplication and division operation.

 Example 3

1, -3, 9, -27, 81, -243,… each number is obtained by multiplying -3 with the previous number.

Exercises

1. Fill the blank spaces using the following pattern: 164, 155, 146, ____, 128, ___, 

2. Identify the type of pattern for the sequence: 0, 3, 6, 9, 12, …

3. Give one example of a number pattern.

4. Identify the type of pattern for the sequence: 1, 1/3, 1/9, 1/27…

1.1.2 Cartesian coordinate system in two dimensions

• The Cartesian coordinate system in two dimensions (also called a rectangular coordinate system) is defined by an ordered pair of perpendicular lines (axes).
• The point where the axes meet is called the origin.
• The points on the Cartesian coordinate system are represented by the two coordinates, like P(x, y).
• From the point P(x, y) the first coordinate is called abscissa and second coordinate is called ordinate of P, respectively.

• The coordinate plane is divided into 4 parts, namely Quadrant I, II, III, and IV as shown in the figure:

Example 4

Plot the points whose coordinates are given on a Cartesian coordinate system. A (1, 1), B(-1, 2), C(2, -1).

Solution:

1.1.3 Basic concepts of relations

Definition:

A relation is a set of ordered pairs. That is the relation is the set whose elements are given by ordered pairs. It is denoted by 𝑅.

Note:

1. A relation establishes pairing between objects.

2. Let A and B be non-empty sets. A relation R from A to B is any subset of A × B.

In other words, R is a relation from A to B if and only if R⊆ (A × B).

Example 5

Let 𝑅 be a relation of the set of all ordered pairs (x, y) of natural numbers where x is a factor of y, then which of the following ordered pairs belongs to 𝑅? (2, 9), (2, 4), (4, 3), (3, 9), (18, −3), (9, 3), (7, 7), (3, 12), (4, 12), (6, 18), (30, 5)

: (2, 4), (3, 9), (7, 7), (3, 12), (4, 12) and (6, 18) belong to the relation R.

Example 6

Let A = {2, 3} and B = {1, 3, 5}

a. R = {(2, 1), (3, 1)} is a relation from A to B. Express the relation using builder method.

b. Is R = {(2, 1), (2, 3), (2, 5), (3, 1), (3, 4), (3, 5)} a relation from A to B? Give the reason for your answer.

c. If R is a relation from A to B given by y = x + 1 where x ∈ A and y∈ B.

Solution:

a. R = {(x, y): x is greater than y; x, y ∈ A}.

b. No. Because in a relation from A to B all the second coordinates must be from set B. For example (3, 4) ∈ R but 4 ∉ B.

c. R = {(2, 3)}.

Domain and Range of a relation

Definition:

Let R be a relation from a set A to a set B. Then:

i. Domain of R = {x: (x, y) belongs to R for some y}.

ii. Range of R = {y: (x, y) belongs to R for some x}.

Example 7

Given the relation R = {(2, 3), (1, 5), (0, 5), (6, 8)}, find the domain and range of the relation R.

Solution:

Since the domain contains the first coordinates, domain = {2, 1, 0, 6} and the range contains the second coordinates, range = {3, 5, 8}

Example 8

Given A = {5, 12, 7, 9, 8, 3} and B = {1, 2, 4, 6, 7}. Find the domain and range of the relation R = {(x, y): x ∈ A, y ∈ B, x < y}.

Solution:

If we describe R by complete listing method, we will find:

R = {(5, 6), (5, 7), (3, 4), (3, 6), (3, 7)}.

This shows that the domain of R = {3, 5} and the range of R = {4, 6, 7}.

1.1.4 Graphs of Relations

Example 9

Sketch the graph of the relation R if R is the set of ordered pairs (x, y) of real numbers and such that y = x.

Solution

We take the values of x, calculate the corresponding values of y, plot the resulting points (x, y) and connect the points.

In general, to sketch graphs of relations involving inequalities, do the following steps

1. Draw the graph of the equation on the -coordinate system.

2. If the relating inequality is ≤ 𝑜𝑟 ≥, use a solid line; if it is < 𝑜𝑟 >, use a broken line.

3. Then take arbitrary ordered pairs represented by the points.

4. The region that contains these points representing the ordered pair satisfying the relation will be the graph of the relation.

Example 10

Sketch the graph of the relation R if R be the set of ordered pairs (x, y) of real numbers and such that y > x.

Solution:

To sketch the graph

1. Draw the line y = x.
2. Since the relation involves  y ≥ x, use the broken line: “the points on the line y = x are not included.”
3. Take points representing ordered pairs, say (0, 3) and (0, −2) from above and below the line y = x.
4. The ordered pair (0, 3) satisfies the relation. Hence, points above the line y = x are members of the relation 𝑅.

Example 11

Sketch the graph of the relation R = {(x, y): y ≥ x + 1; x ∈ ℝ and y ∈ ℝ}.

Solution:

1.  Draw the graph of the line y = x + 1.
2. Since the relating inequality is use solid line
3. Select two points with coordinates (0, 5) and (2, 0). Obviously (0, 5) satisfies the relation.
4. Shade the region containing the point with coordinates (0, 5) So the graph of the relation R = {(x, y): y ≥x + 1} is as shown by the shaded region.

Example 12

Sketch the graph of the relation R={(x, y): y ≥ x+2 and y >-x, x ∈ R; and y ∈ R}.

Solution:

First sketch the graph of the relation

R={(x, y): y ≥ x+2, x ∈ R; and y ∈R}

Next, on the same diagram, sketch the graph of R={(x, y): y >-x, x ∈R and y ∈R}.

The two shaded regions have some overlap. The Intersection of the two regions is the graph of the relation. So, taking only the common region, we obtain the graph of the relation.

The graph of R={(x, y): y ≥ x+2 and y >-x, x ∈ R and y ∈R} as shown in Figure below.