Key Concepts and Formulas

• Equivalence Relation Properties: For a relation $R$ on a set $S$ to be an equivalence relation, it must be reflexive ($a R a$), symmetric (if $a R b$ then $b R a$), and transitive (if $a R b$ and $b R c$ then $a R c$).
• Equivalence Class: The equivalence class of an element $x$ is the set of all elements $y$ such that $x R y$.
• Ratio Representation: The condition $(a, b) \simeq (c, d)$ if and only if $ad = bc$ is equivalent to $\frac{a}{b} = \frac{c}{d}$ (assuming $b, d \neq 0$). This means ordered pairs are in the same equivalence class if they represent the same ratio.

Step-by-Step Solution

Step 1: Understand the Equivalence Relation The given equivalence relation is $(a, b) \simeq (c, d)$ if and only if $ad = bc$. We are given the set $A = \{2, 3, 4, \dots, 30\}$. We need to find the number of ordered pairs $(c, d)$ from $A \times A$ that satisfy $(c, d) \simeq (4, 3)$.

Explanation: This step defines the problem: we are looking for elements in the equivalence class of $(4, 3)$ within the set $A \times A$. The condition $ad = bc$ connects the components of the ordered pairs.

Step 2: Translate the Equivalence Condition for (4, 3) We are looking for pairs $(c, d)$ such that $(c, d) \simeq (4, 3)$. Using the definition of the relation with $a=c, b=d$ and $c=4, d=3$ from the given pair, we get: $c \cdot 3 = d \cdot 4$ $3c = 4d$

Explanation: This step applies the given equivalence relation to the specific pair $(4, 3)$ and the unknown pair $(c, d)$. We substitute the values into the defining equation.

Step 3: Express the Relationship as a Ratio From the equation $3c = 4d$, we can rearrange it to find the ratio between $c$ and $d$. Assuming $d \neq 0$ (which is true since elements of $A$ are positive integers), we can write: $\frac{c}{d} = \frac{4}{3}$

Explanation: This step highlights the core property of the equivalence relation. All pairs $(c, d)$ equivalent to $(4, 3)$ must have their components in the ratio $4:3$.

Step 4: Represent $c$ and $d$ using a Common Factor Since $c$ and $d$ must be integers from set $A$, and their ratio is $\frac{4}{3}$ (which is in its simplest form), we can express $c$ and $d$ as multiples of a common integer factor $k$. Let: $c = 4k$ $d = 3k$ where $k$ is a positive integer.

Explanation: This is a standard method for finding integer solutions to ratio problems. Since $\frac{c}{d} = \frac{4}{3}$, $c$ must be a multiple of 4 and $d$ must be the corresponding multiple of 3.

Step 5: Apply the Constraints from Set A The elements $c$ and $d$ must belong to the set $A = \{2, 3, 4, \dots, 30\}$. This imposes the following constraints: $2 \le c \le 30$ $2 \le d \le 30$

Explanation: This is a critical step where we incorporate the domain of the elements of the ordered pairs. Both components of the ordered pair must be valid members of set $A$.

Step 6: Determine the Possible Values of $k$ Substitute $c = 4k$ and $d = 3k$ into the inequalities from Step 5:

For $c$: $2 \le 4k \le 30$ Dividing by 4: $\frac{2}{4} \le k \le \frac{30}{4}$ $0.5 \le k \le 7.5$

For $d$: $2 \le 3k \le 30$ Dividing by 3: $\frac{2}{3} \le k \le \frac{30}{3}$ $0.66... \le k \le 10$

We need to find the integer values of $k$ that satisfy both ranges. The combined range for $k$ is: $\max(0.5, 0.66...) \le k \le \min(7.5, 10)$ $0.66... \le k \le 7.5$

Since $k$ must be a positive integer, the possible values for $k$ are $\{1, 2, 3, 4, 5, 6, 7\}$.

Explanation: We solve for the possible range of the common factor $k$ by considering the constraints on both $c$ and $d$. The intersection of these ranges gives us the valid values for $k$. Since $c$ and $d$ must be integers, $k$ must also be an integer.

Step 7: List the Valid Ordered Pairs For each valid integer value of $k$, we find the corresponding ordered pair $(c, d) = (4k, 3k)$:

• $k=1 \implies (c, d) = (4 \cdot 1, 3 \cdot 1) = (4, 3)$
• $k=2 \implies (c, d) = (4 \cdot 2, 3 \cdot 2) = (8, 6)$
• $k=3 \implies (c, d) = (4 \cdot 3, 3 \cdot 3) = (12, 9)$
• $k=4 \implies (c, d) = (4 \cdot 4, 3 \cdot 4) = (16, 12)$
• $k=5 \implies (c, d) = (4 \cdot 5, 3 \cdot 5) = (20, 15)$
• $k=6 \implies (c, d) = (4 \cdot 6, 3 \cdot 6) = (24, 18)$
• $k=7 \implies (c, d) = (4 \cdot 7, 3 \cdot 7) = (28, 21)$

All these pairs have both components within the set $A = \{2, 3, \dots, 30\}$.

Explanation: By substituting each valid integer value of $k$ back into the expressions for $c$ and $d$, we generate all the ordered pairs that satisfy the given equivalence relation and the set constraints.

Step 8: Count the Number of Ordered Pairs We have found 7 possible integer values for $k$. Each value of $k$ corresponds to a unique ordered pair $(c, d)$ that satisfies the given conditions. Therefore, there are 7 such ordered pairs.

Explanation: The number of valid integer values for $k$ directly translates to the number of ordered pairs in the equivalence class.

Common Mistakes & Tips

• Ignoring Set Constraints: Always verify that both elements of the ordered pair $(c, d)$ fall within the specified set $A$. Forgetting this can lead to an incorrect count.
• Incorrect Range Intersection: When determining the valid range for $k$, ensure you correctly find the maximum of the lower bounds and the minimum of the upper bounds.
• Simplifying Ratios: If the ratio $a:b$ is not in its simplest form, always simplify it first (e.g., $6:4$ becomes $3:2$) before setting up $c=3k, d=2k$.

Summary

The problem asks for the number of ordered pairs $(c, d)$ from $A \times A$ that are equivalent to $(4, 3)$ under the relation $(a, b) \simeq (c, d) \iff ad = bc$. This relation implies that the ratio $\frac{a}{b}$ must equal $\frac{c}{d}$. For $(4, 3)$, this means $\frac{c}{d} = \frac{4}{3}$. We express $c=4k$ and $d=3k$. By applying the constraints $2 \le c \le 30$ and $2 \le d \le 30$, we found that $k$ can be any integer from 1 to 7, inclusive. This gives a total of 7 ordered pairs.

The final answer is $\boxed{7}$.