Let W = \{1, 2, 3, 4\} and R 1 = {(1, 2), (4, 3), (2, 2), (2, 1), (3, 1)\}  be the relation definition on W. Verify R={1} equivalence or not.

b) Let X = \{1, 2, 3, 4\} Y = \{5, 7, 9, 11, 12\} and f: X→Y defined by f(x) = 2x + 3 Represent f as

1) Set of ordered pairs. il) an arrow diagram.

### Part a) Verifying \( R_1 \) as an equivalence relation

To determine whether \( R_1 \) is an equivalence relation on the set \( W = \{1, 2, 3, 4\} \), we need to check whether it satisfies the following three properties of an equivalence relation:

1. **Reflexivity**: For every element \( x \in W \), the pair \( (x, x) \) should be in \( R_1 \).
2. **Symmetry**: For every pair \( (a, b) \in R_1 \), if \( (a, b) \in R_1 \), then \( (b, a) \) must also be in \( R_1 \).
3. **Transitivity**: If \( (a, b) \in R_1 \) and \( (b, c) \in R_1 \), then \( (a, c) \) must also be in \( R_1 \).

#### 1. Reflexivity:
We check if \( (x, x) \in R_1 \) for all \( x \in W \).
- \( (1, 1) \notin R_1 \)
- \( (2, 2) \in R_1 \) (this satisfies reflexivity for 2)
- \( (3, 3) \notin R_1 \)
- \( (4, 4) \notin R_1 \)

Since reflexivity is not satisfied for all elements of \( W \), **\( R_1 \) is not an equivalence relation**. Reflexivity alone is violated, so there's no need to check the other properties (symmetry and transitivity).

### Part b) Function \( f: X \to Y \) defined by \( f(x) = 2x + 3 \)

Given:
- \( X = \{1, 2, 3, 4\} \)
- \( Y = \{5, 7, 9, 11, 12\} \)
- The function is defined as \( f(x) = 2x + 3 \).

#### 1. **Set of ordered pairs representation of \( f \)**:
We evaluate \( f(x) \) for each \( x \in X \):
- \( f(1) = 2(1) + 3 = 5 \)
- \( f(2) = 2(2) + 3 = 7 \)
- \( f(3) = 2(3) + 3 = 9 \)
- \( f(4) = 2(4) + 3 = 11 \)

Thus, the set of ordered pairs is:
\[
\{(1, 5), (2, 7), (3, 9), (4, 11)\}
\]

#### 2. **Arrow diagram**:
In the arrow diagram, we represent elements of \( X \) as nodes on the left, elements of \( Y \) as nodes on the right, and draw arrows from \( x \in X \) to \( f(x) \in Y \).

Here's the correspondence based on the function \( f \):
- \( 1 \to 5 \)
- \( 2 \to 7 \)
- \( 3 \to 9 \)
- \( 4 \to 11 \)

This forms a simple mapping from \( X \) to \( Y \).

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Do you want further details or have any questions?

**5 related questions**:
1. How do we formally prove that a relation is symmetric?
2. Can a relation still be transitive if it fails reflexivity or symmetry?
3. How can we extend the function \( f \) if more elements are added to \( X \) or \( Y \)?
4. What would happen if the function \( f(x) = 2x + 3 \) mapped to values outside \( Y \)?
5. What are some real-world applications of equivalence relations and functions?

**Tip**: When verifying if a relation is an equivalence relation, failing any one of reflexivity, symmetry, or transitivity is enough to disqualify it.

Let W = {1, 2, 3, 4} and R1 = {(1, 2), (4, 3), (2, 2), (2, 1), (3, 1)} be the relation definition on W. Verify if R1 is an equivalence relation or not. b) Let X = {1, 2, 3, 4}, Y = {5, 7, 9, 11, 12}, and f: X → Y defined by f(x) = 2x + 3. Represent f as 1) Set of ordered pairs. 2) An arrow diagram.

Explore how to verify whether a given relation is an equivalence relation using properties such as reflexivity, symmetry, and transitivity. Also, learn to represent the function f(x) = 2x + 3 as a set of ordered pairs and an arrow diagram, suitable for students in Grades 9-12.

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