Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not? (i) $R = \{(2, 1), (3, 1), (4, 2)\}$ (ii) $R = \{(2, 2), (2, 4), (3, 3), (4, 4)\}$ (iii) $R = \{(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)\}$.

Question

Accepted Answer

> [!TLDR]
> A relation is a function if every input (first element) has exactly one output (second element).  
> (i) **Yes**, each input maps to a unique output.  
> (ii) **No**, input 2 maps to two different outputs (2 and 4).  
> (iii) **Yes**, each input maps to exactly one output.

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### The Core Idea: The Arrow Diagram Test

Think of a relation as a set of arrows from one number (the input) to another (the output). For it to be a **function**, every input must have **exactly one arrow** leaving it. If any input has two or more arrows (two different outputs), or if an input is missing entirely, it’s not a function — but missing inputs are fine; only the inputs that *appear* must be unambiguous.

Let’s apply this to each case.

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#### (i) $R = \{(2, 1), (3, 1), (4, 2)\}$

1. **List the inputs (first elements):** 2, 3, 4.  
2. **Check each input:**  
   - 2 → only 1  
   - 3 → only 1  
   - 4 → only 2  
3. No input repeats with a different output. Every input that appears has a single, unique output.

> [!NOTE]
> It’s perfectly fine that both 2 and 3 map to the same output (1). A function can have many inputs sharing the same output — that’s called a many-to-one mapping, and it’s allowed.

**Conclusion:** This is a function.

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#### (ii) $R = \{(2, 2), (2, 4), (3, 3), (4, 4)\}$

1. **List the inputs:** 2, 2, 3, 4.  
2. **Check each input:**  
   - 2 appears twice: once with output 2, once with output 4.  
   - 3 → only 3  
   - 4 → only 4

> [!WARNING]
> The input 2 has **two different arrows** leaving it (to 2 and to 4). This violates the definition of a function. Even if one of those pairs were removed, it would become a function — but as given, it’s not. …

Q.Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not?

Concept understanding — Testing Whether a Relation Is a Function

The Core Idea: The Arrow Diagram Test

(i) $R = \{(2, 1), (3, 1), (4, 2)\}$

(ii) $R = \{(2, 2), (2, 4), (3, 3), (4, 4)\}$

Q.Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not?

The Core Idea: The Arrow Diagram Test

(i) R={(2,1),(3,1),(4,2)}R = \{(2, 1), (3, 1), (4, 2)\}R={(2,1),(3,1),(4,2)}

(ii) R={(2,2),(2,4),(3,3),(4,4)}R = \{(2, 2), (2, 4), (3, 3), (4, 4)\}R={(2,2),(2,4),(3,3),(4,4)}

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(i) $R = \{(2, 1), (3, 1), (4, 2)\}$

(ii) $R = \{(2, 2), (2, 4), (3, 3), (4, 4)\}$

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