Question

What are the sets in the partition of the integers arising from
congruence modulo 4

Answer 1

Integers are partitioned into four equivalence classes based on congruence modulo 4: integers congruent to 0, 1, 2, or 3 modulo 4. Each class contains all integers with the same remainder when divided by 4.

Step-by-step explanation:

The question is about the partition of the integers when divided into equivalence classes according to congruence modulo 4.

A partition of a set breaks it down into distinct subsets called equivalence classes such that every element of the original set is in exactly one of these subsets.

In modular arithmetic, two integers are considered congruent modulo 4 if they have the same remainder when divided by 4.

The equivalence classes for congruence modulo 4 are therefore:

0 modulo 4: {..., -8, -4, 0, 4, 8, ...}

1 modulo 4: {..., -7, -3, 1, 5, 9, ...}

2 modulo 4: {..., -6, -2, 2, 6, 10, ...}

3 modulo 4: {..., -5, -1, 3, 7, 11, ...}

These four equivalence classes form a partition of the integers under the relation of congruence modulo 4.

No integer can be in more than one class, fulfilling the definition of a partition.

Answer 2

In congruence modulo 4, the integers can be divided into four sets based on their remainder when divided by 4. These four sets form a partition of the integers.

Step-by-step explanation:

In congruence modulo 4, the integers can be divided into four sets based on their remainder when divided by 4:

1. All integers that leave a remainder of 0 when divided by 4, such as 4, 8, and -12.
2. All integers that leave a remainder of 1 when divided by 4, such as 1, 5, and -15.
3. All integers that leave a remainder of 2 when divided by 4, such as 2, 6, and -10.
4. All integers that leave a remainder of 3 when divided by 4, such as 3, 7, and -13.

These four sets, when combined, form a partition of the integers based on congruence modulo 4.