What is the Range of a Relation?

S3-SA1-0499

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Range of a Relation is the set of all the second elements (or output values) in the ordered pairs that make up the relation. It tells us all the possible 'results' or 'outcomes' we can get from that relation.

Simple Example

Quick Example

Imagine a shop selling samosas. If the relation shows (Number of Samosas, Total Cost), like {(1, 15), (2, 30), (3, 45)}. Here, the costs (15, 30, 45) form the Range. It's all the different amounts you might pay.

Worked Example

Step-by-Step

Let's find the Range of the relation R = {(Cricket Player, Runs Scored)} where R = {(Virat Kohli, 80), (Rohit Sharma, 55), (KL Rahul, 80), (Hardik Pandya, 30)}.

Step 1: Identify the relation as a set of ordered pairs.
---Step 2: Recall that the Range consists of all the second elements (output values) of these ordered pairs.
---Step 3: List all the second elements from the given relation: 80, 55, 80, 30.
---Step 4: Form a set using these second elements. Remember, in a set, we list each unique element only once.
---Step 5: The unique second elements are 80, 55, 30.
---Step 6: Write these unique elements as a set.
---Answer: The Range of the relation R is {30, 55, 80}.

Why It Matters

Understanding Range is crucial in fields like Data Science and AI, where you analyze data patterns and predict possible outcomes. Engineers use it to know the limits of what a system can produce, and in Economics, it helps analyze possible price changes or production levels.

Common Mistakes

MISTAKE: Including the first elements (input values) in the Range. | CORRECTION: The Range only includes the second elements of the ordered pairs.

MISTAKE: Listing repeated elements multiple times in the Range. | CORRECTION: When writing a set for the Range, each unique element should be listed only once.

MISTAKE: Confusing Range with Domain. | CORRECTION: The Domain is the set of first elements, while the Range is the set of second elements.

Practice Questions

Try It Yourself

QUESTION: What is the Range of the relation M = {(Monday, 25°C), (Tuesday, 28°C), (Wednesday, 25°C), (Thursday, 30°C)}? | ANSWER: {25°C, 28°C, 30°C}

QUESTION: Find the Range of the relation P = {(x, y) | y = x + 2, where x is from {1, 2, 3}}. | ANSWER: {3, 4, 5}

QUESTION: A mobile data plan offers (GB used, Cost in Rupees). If the relation is {(1, 100), (2, 180), (3, 250), (4, 300), (5, 300)}. What is the Range of this relation? | ANSWER: {100, 180, 250, 300}

MCQ

Quick Quiz

For the relation R = {(Pen, 10), (Pencil, 5), (Eraser, 5), (Book, 50)}, what is the Range?

{Pen, Pencil, Eraser, Book}

{10, 5, 50}

{5, 10, 50}

{10, 5, 5, 50}

The Correct Answer Is:

The Range is the set of all unique second elements. The second elements are 10, 5, 5, 50. When listed uniquely in a set, they become {5, 10, 50}.

Real World Connection

In the Real World

Think about online food delivery apps like Zomato or Swiggy. If a relation shows (Restaurant, Average Delivery Time in minutes), the Range would be all the different possible delivery times you might see. It helps them understand how fast they can deliver across different restaurants.

Key Vocabulary

Key Terms

RELATION: A set of ordered pairs showing a connection between two sets. | ORDERED PAIR: A pair of elements (a, b) where the order matters. | DOMAIN: The set of all first elements in a relation. | SET: A collection of distinct objects.

What's Next

What to Learn Next

Great job understanding the Range! Next, you should explore 'Types of Relations' like one-to-one or many-to-one. This will help you understand how different inputs connect to outputs in more specific ways, which is super useful for advanced math.