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.