Solved problems on defining relations using ordered pairs

📚 What is a Relation?

In mathematics, a relation describes a connection between two sets. Think of it as a rule that links elements from one set (the domain) to elements of another set (the codomain). We often represent these connections using ordered pairs.

• 🔍 Ordered Pair: An ordered pair, typically written as $(x, y)$, represents a specific relationship between $x$ and $y$. The order matters – $(x, y)$ is generally different from $(y, x)$.
• 📝 Defining a Relation: A relation is formally defined as a set of ordered pairs. Each ordered pair $(x, y)$ indicates that $x$ is related to $y$ in some specific way.

📜 Historical Context

The concept of relations has been fundamental to the development of mathematics since the formalization of set theory in the late 19th century. Mathematicians like Georg Cantor and Richard Dedekind laid the groundwork for understanding relationships between mathematical objects, leading to the modern definition of relations as sets of ordered pairs. This formalization enabled precise and rigorous analysis of mathematical structures.

⭐ Key Principles

• 🔗 Domain: The domain of a relation is the set of all first elements (x-values) in the ordered pairs.
• 🎯 Range: The range of a relation is the set of all second elements (y-values) in the ordered pairs.
• 🗺️ Representation: Relations can be represented in several ways, including:
  • 📈 As a set of ordered pairs: { (1, 2), (3, 4), (5, 6) }
  • 📊 Graphically: Plotting the ordered pairs on a coordinate plane.
  • ✍️ Using a table: Listing x and y values in columns.
  • 🖋️Using an equation: An equation that defines the relationship between x and y, like $y = x + 1$.

🧮 Solved Problems

Problem 1: Identifying a Relation

Given the set A = {1, 2, 3} and B = {a, b}, is the following set of ordered pairs a relation from A to B? R = { (1, a), (2, b), (3, a) }

Solution:

Yes, R is a relation from A to B because each ordered pair in R has its first element from A and its second element from B.

Problem 2: Finding the Domain and Range

Find the domain and range of the relation R = { (1, 4), (2, 5), (3, 6) }.

Solution:

• 🏠 Domain = {1, 2, 3}
• 🏞️ Range = {4, 5, 6}

Problem 3: Relation Defined by an Equation

Given the equation $y = x^2$, and the set A = {-2, -1, 0, 1, 2}, express the relation as a set of ordered pairs.

Solution:

• 🧪For $x = -2$, $y = (-2)^2 = 4$, so the ordered pair is (-2, 4).
• 💡For $x = -1$, $y = (-1)^2 = 1$, so the ordered pair is (-1, 1).
• ⚛️ For $x = 0$, $y = (0)^2 = 0$, so the ordered pair is (0, 0).
• 📈 For $x = 1$, $y = (1)^2 = 1$, so the ordered pair is (1, 1).
• 🔬 For $x = 2$, $y = (2)^2 = 4$, so the ordered pair is (2, 4).

Therefore, the relation is { (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4) }.

Problem 4: Determining if a Relation is a Function

Is the relation R = { (1, a), (2, b), (3, a) } a function? A = {1, 2, 3} and B = {a,b}

Solution:

Yes, R is a function because each element in the domain (A) is associated with exactly one element in the codomain (B). Even though 'a' is associated with both 1 and 3, each element in A only has one corresponding element.

Problem 5: Relation with restrictions.

Express the relation R = {(x, y) | y = x + 2, where x ∈ {1, 2, 3, 4}} as a set of ordered pairs.

Solution:

• 🔑 When $x = 1$, $y = 1 + 2 = 3$. So, (1, 3) is in R.
• 🔩 When $x = 2$, $y = 2 + 2 = 4$. So, (2, 4) is in R.
• 📐 When $x = 3$, $y = 3 + 2 = 5$. So, (3, 5) is in R.
• 💡 When $x = 4$, $y = 4 + 2 = 6$. So, (4, 6) is in R.

Therefore, R = {(1, 3), (2, 4), (3, 5), (4, 6)}.

Problem 6: Representing Relations in a Table

Represent the relation R = {(1, 5), (2, 6), (3, 7), (4, 8)} in a table format.

Solution:

Problem 7: Graphical representation.

Plot the relation R = {(-1, 2), (0, 0), (1, 2), (2, 4)} on a coordinate plane.

Solution:

The relation is plotted by marking each of the given ordered pairs as points on a coordinate plane. (-1,2) would be in quadrant II, (0,0) would be the origin, (1,2) would be in quadrant I and (2,4) would also be in quadrant I.

📝 Practice Quiz

Test your knowledge with these practice questions!

1. If A = {a, b, c} and B = {1, 2}, is R = {(a, 1), (b, 2), (c, 1)} a relation from A to B?
2. What is the domain and range of R = {(4, 7), (5, 8), (6, 9)}?
3. Express the relation defined by $y = 2x - 1$ for $x$ ∈ {0, 1, 2, 3} as a set of ordered pairs.
4. Is the relation R = {(1, 2), (1, 3), (2, 4)} a function?
5. Represent the relation R = {(5, 10), (6, 12), (7, 14)} in a table format.
6. Given the relation R={(x, y) | y > x, x ∈ {1, 2, 3, 4} and y ∈ {4, 5, 6, 7}}, determine R as a set of ordered pairs.
7. Plot the relation R = {(-2, 4), (-1, 1), (0, 0), (1, 1)} on a coordinate plane.

💡 Conclusion

Understanding relations using ordered pairs is crucial in mathematics. By grasping the definitions of domain, range, and different representations, you can easily solve problems involving relations. Keep practicing, and you'll master this concept in no time!