# Equivalence Classes Homework: (a) & (b)

• PolyFX
In summary, equivalence classes are sets of elements that are considered equivalent based on some defined criteria. They are often used in homework problems to simplify complex problems and group together elements that share similar properties. An example of using equivalence classes in a homework problem would be dividing a set of numbers into groups based on their divisibility by 3. Two elements belong in the same equivalence class if they share the same properties defined in the problem, and equivalence classes can be used for any type of elements, not just numerical values.
PolyFX

## Homework Statement

$$\forall (a,b), (c,d) \in (Z^2), (a,b)D(c,d) \leftrightarrow a\equiv c\mod\2\and\b\equiv d mod 3$$

*edit* Sorry the b = d mod 3 is all part of the same line.

(a) List four elements of the equivalence class [{5,3}]

(b) How many equivalence classes of D are there in total? List a representative element of each of them.

## The Attempt at a Solution

(5,3)D(c,d)

a = c mod 2 can also be written as 2 = a-c
b = d mod 3 can also be written as 3 = b-d

Would I need to somehow use the above two lines in this problem?

I'm very lost on this one and have no clue where to start.

I took the liberty of cleaning up your LaTeX formatting to make everything visible.
PolyFX said:

## Homework Statement

$$\forall (a,b), (c,d) \in (Z^2), (a,b) D (c,d) \leftrightarrow a \equiv c~mod~2~and~b~\equiv~ d~mod~3$$

*edit* Sorry the b = d mod 3 is all part of the same line.

(a) List four elements of the equivalence class [{5,3}]

(b) How many equivalence classes of D are there in total? List a representative element of each of them.

## The Attempt at a Solution

(5,3)D(c,d)

a = c mod 2 can also be written as 2 = a-c
b = d mod 3 can also be written as 3 = b-d

Would I need to somehow use the above two lines in this problem?

I'm very lost on this one and have no clue where to start.

In mod 2 arithmetic, every integer is congruent to either 0 or 1. In mod 3 arithmetic, every integer is congruent to 0, 1, or 2. For a pair of numbers, this represents 6 possibilities.

So "a mod b" is always going to produce an integer "b - 1"?

So for part a here is what I have so far;

a == c mod 2
=> a mod 2 = c mod 2

we let a = 5

so 5 mod 2 = c mod 2

Is this correct so far?

PolyFX said:

So "a mod b" is always going to produce an integer "b - 1"?
I don't know what you mean by "b - 1". a mod b will be one of b integer values: 0, 1, 2, 3, ..., b - 1. a mod b is the remainder after a is divided by b.
PolyFX said:
So for part a here is what I have so far;

a == c mod 2
=> a mod 2 = c mod 2

we let a = 5

so 5 mod 2 = c mod 2

Is this correct so far?
Yes, but you're not really getting all that far. 5 mod 2 = 1, 7 mod 2 = 1, 6 mod 2 = 0. In mod 2 arithmetic, every integer is put into one of two buckets: all the odd integers go in one bucket, and all the even integers go in the other bucket.

In mod 3 arithmetic, all integers go into one of three buckets. 3, 6, 9, ... go into one bucket. 4, 7, 10, ... go into another bucket, and 5, 8, 11, ... go into the third bucket. Each number in the first group here is congruent to 0 mod 3. Each number in the second group is congruent to 1 mod 3, and each number in the third group is congruent to 2 mod 3.
PolyFX said:

Oh I see now.

so

5 mod 2 = 1
c mod 2 = 1

therefore c can be 5, 7, 9, 11, etc. since 7/2 has a remainder of 1 and 9/2 has a remainder of one.

Similarly,

3 mod 3 = 0 so;
d mod 3 = 0

therefore d can be 3, 6, 9, 12 etc.

So would 4 ordered pairs be;

(5,3), (7,6), (9,9), (11,12)?

-Thanks again.

Yes, those ordered pairs would be in the same equivalence class as (5, 3), as would (1, 0).

To clarify a couple of things you wrote, if c mod 2 = 1, then c could be any odd integer, including the negative ones. All of them could be represented as {..., -3, -1, 1, 3, 5, ...}

If d mod 3 = 0, then d is any integer that is evenly divisibly by 3. All of them would be {..., -6, -3, 0, 3, 6, 9, 12, ...}

## 1. What are equivalence classes?

Equivalence classes are sets of elements that are considered equivalent based on some defined criteria. In other words, elements within an equivalence class share similar properties or characteristics.

## 2. How are equivalence classes used in homework problems?

In homework problems, equivalence classes are often used to group together elements that are considered equivalent. This approach can help simplify complex problems and make them easier to solve.

## 3. Can you provide an example of using equivalence classes in a homework problem?

Sure! Let's say you have a homework problem where you need to divide a set of numbers into groups based on their divisibility by 3. The equivalence classes in this case would be {3, 6, 9, ...}, {1, 4, 7, ...}, and {2, 5, 8, ...} where each set contains numbers that have a remainder of 0, 1, or 2 when divided by 3, respectively.

## 4. How do you determine if two elements belong in the same equivalence class?

Two elements belong in the same equivalence class if they share the same properties or characteristics defined in the problem. This can include being divisible by the same number, having the same color, or any other defined criteria.

## 5. Are equivalence classes limited to numerical values?

No, equivalence classes can be used for any type of elements, including numerical values, strings, objects, and more. As long as the elements share similar properties or characteristics based on the defined criteria, they can be grouped into an equivalence class.

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