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7. Consider A = {1, 2, 3, 4, 5, 6}. Let R be the equivalence relation on A defined by: $R = \{ (1,1), (1,5), (2,2), (3,3), (4,4), (5,5), (6,6) \}$ Find the partition of A induced by $R$.

This problem is asking for the equivalence classes based on the relation $R$. To solve, we examine the pairs and find how elements are grouped by equivalence.

Explanation:

The relation $R$ tells us which elements are related to each other, and equivalence relations induce partitions on sets.

Given:

• $(1,1), (1,5), (5,5)$: Elements 1 and 5 are related, so they belong to the same equivalence class.
• $(2,2)$: Element 2 is related only to itself, so it forms its own equivalence class.
• $(3,3)$: Element 3 is related only to itself.
• $(4,4)$: Element 4 is related only to itself.
• $(6,6)$: Element 6 is related only to itself.

Partition:

• $\{1, 5\}$
• $\{2\}$
• $\{3\}$
• $\{4\}$
• $\{6\}$

8. Let S be the set of integers from 1 to 12. In S, define a relation as $x \sim y$ if the product of the digits in $x$ is the same as the product of the digits in $y$. Find the number of members in the partition of S induced by this relation and list them.

Explanation:

Here we have a relation based on the product of the digits in each number in the set $S = \{1, 2, 3, \dots, 12\}$.

1. For single-digit numbers, the product of digits is just the number itself:

   • $1, 2, 3, 4, 5, 6, 7, 8, 9$.

2. For two-digit numbers:

   • $10$ has a product of $1 \times 0 = 0$.
   • $11$ has a product of $1 \times 1 = 1$.
   • $12$ has a product of $1 \times 2 = 2$.

Now we group elements with the same product of digits:

• Product $0$: $\{10\}$.
• Product $1$: $\{1, 11\}$.
• Product $2$: $\{2, 12\}$.
• Product $3$: $\{3\}$.
• Product $4$: $\{4\}$.
• Product $5$: $\{5\}$.
• Product $6$: $\{6\}$.
• Product $7$: $\{7\}$.
• Product $8$: $\{8\}$.
• Product $9$: $\{9\}$.

Number of partitions:

• $10$ equivalence classes: $\{10\}, \{1, 11\}, \{2, 12\}, \{3\}, \{4\}, \{5\}, \{6\}, \{7\}, \{8\}, \{9\}$

Do you need details or further clarification on any part of the solution?

Here are 5 related questions to practice:

1. What is an equivalence relation, and how is it defined?
2. How do equivalence relations induce partitions in sets?
3. Given a set and an equivalence relation, how can you find the equivalence classes?
4. If you change the relation $R$ in Problem 7, how does it affect the partition?
5. Can you think of other real-world situations where equivalence relations are useful?

Tip: When working with equivalence relations, always check for the properties of reflexivity, symmetry, and transitivity to confirm the relation!