To determine the equivalence classes under the relation "≡ mod 3" for the set $\{1, 2, 3, 4, 5, 6, 7\}$, we group elements that are congruent to each other modulo 3. Two numbers $a$ and $b$ are in the same equivalence class if and only if $a \equiv b \pmod{3}$, meaning the remainder when $a$ and $b$ are divided by 3 is the same.

Steps:

1. Compute the remainder when each element of the set is divided by 3.
2. Group elements with the same remainder.

Calculation:

• $1 \div 3 = 0 \text{ remainder } 1 \implies 1 \equiv 1 \pmod{3}$
• $2 \div 3 = 0 \text{ remainder } 2 \implies 2 \equiv 2 \pmod{3}$
• $3 \div 3 = 1 \text{ remainder } 0 \implies 3 \equiv 0 \pmod{3}$
• $4 \div 3 = 1 \text{ remainder } 1 \implies 4 \equiv 1 \pmod{3}$
• $5 \div 3 = 1 \text{ remainder } 2 \implies 5 \equiv 2 \pmod{3}$
• $6 \div 3 = 2 \text{ remainder } 0 \implies 6 \equiv 0 \pmod{3}$
• $7 \div 3 = 2 \text{ remainder } 1 \implies 7 \equiv 1 \pmod{3}$

Grouping by Remainders:

• Numbers with remainder $0$: $\{3, 6\}$
• Numbers with remainder $1$: $\{1, 4, 7\}$
• Numbers with remainder $2$: $\{2, 5\}$

Equivalence Classes:

The equivalence classes under the relation $≡ \pmod{3}$ are:

1. $[0] = \{3, 6\}$
2. $[1] = \{1, 4, 7\}$
3. $[2] = \{2, 5\}$

Let me know if you'd like further clarification or have questions!

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Related Questions:

1. How can equivalence relations be visualized using modular arithmetic?
2. What are the properties of equivalence relations and how do they apply here?
3. Can we generalize this equivalence class concept for other moduli, like mod 4 or mod 5?
4. How would the equivalence classes change if the set were expanded, e.g., $\{1, 2, \dots, 10\}$?
5. How can equivalence classes be used in solving systems of congruences?

Tip: Remember that equivalence classes partition the set into disjoint subsets where every element belongs to exactly one class.