# QI. (a) Let A {258, 388, 406, 457, 458, 548, 565, 603, 719, 798, 960} and R be the relation on A defined by R {(x, y) e A x A Ix y(mod 8)).

(i) Find the equivalence class of each element in A. (3 marks)
(ii) Find the quotient set of A. (2 marks)
(iii) Show that the quotient set of A is a partition of A. (2 marks)
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## 5 replies

3 months ago
(a) (i) To find the equivalence class of each element in A, we need to determine all elements in A that are related to each element in A under the relation R.
For example, let's take the element 258:
- The elements in A that are related to 258 under the relation R (mod 8) are those elements whose difference with 258 is divisible by 8.
- So, the equivalence class of 258 would include 258, 386 (258 + 128), and 514 (258 + 256).
Similarly, we can find equivalence classes for each element in A using the same method.
(ii) To find the quotient set of A, we need to list all the distinct equivalence classes obtained in part (i).
(iii) To show that the quotient set of A is a partition of A, we need to demonstrate two things:
- Every element in A belongs to exactly one equivalence class in the quotient set.
- The equivalence classes in the quotient set are pairwise disjoint (meaning they have no elements in common).
Let's compute these step by step:
(i) Equivalence classes:
- Equivalence class of 258: {258, 386, 514}
- Equivalence class of 388: {388, 516, 644}
- Equivalence class of 406: {406, 534, 662}
- Equivalence class of 457: {457, 585, 713}
- Equivalence class of 458: {458, 586, 714}
- Equivalence class of 548: {548, 676, 804}
- Equivalence class of 565: {565, 693, 821}
- Equivalence class of 603: {603, 731, 859}
- Equivalence class of 719: {719, 847, 975}
- Equivalence class of 798: {798, 926, 1054}
- Equivalence class of 960: {960, 1088, 1216}
(ii) Quotient set: The quotient set is the set of all distinct equivalence classes: { {258, 386, 514}, {388, 516, 644}, {406, 534, 662}, {457, 585, 713}, {458, 586, 714}, {548, 676, 804}, {565, 693, 821}, {603, 731, 859}, {719, 847, 975}, {798, 926, 1054}, {960, 1088, 1216} }
(iii) Partition of A:
- Every element in A belongs to exactly one equivalence class in the quotient set.
- Equivalence classes in the quotient set are pairwise disjoint.
Thus, the quotient set of A forms a partition of A.
3 months ago
(i)
To find the equivalence classes of the relation R defined on the set A, where R = {(x, y) ∈ A × A | x ≡ y (mod 8)}, we need to group the elements of A based on their remainders when divided by 8.
The possible remainders when divided by 8 are 0, 1, 2, 3, 4, 5, 6, and 7. Therefore, we will have at most 8 equivalence classes.

Let's consider each remainder and group the elements of A accordingly:
Remainder 0: {960}
Remainder 1: {} (no elements in A)
Remainder 2: {458}
Remainder 3: {603}
Remainder 4: {388, 548}
Remainder 5: {565}
Remainder 6: {406, 718, 798}
Remainder 7: {257, 457}

Therefore, the equivalence classes of the relation R on the set A are:
[0] = {960}
[1] = {} (empty set)
[2] = {458}
[3] = {603}
[4] = {388, 548}
[5] = {565}
[6] = {406, 718, 798}
[7] = {257, 457}
"The notation [x] represents the equivalence class containing the elements that have a remainder of x when divided by 8."

(ii) The quotient set A/R is the set of all distinct equivalence classes. Therefore, the quotient set A/R is:
A/R = {{960}, {}, {458}, {603}, {388, 548}, {565}, {406, 718, 798}, {257, 457}}.

(iii)
1. Union of all sets in A/R is equal to A: The union of all equivalence classes in A/R covers all elements of A.
2. All sets in A/R are non-empty (except possibly the empty set): Each non-empty equivalence class contains elements from A.
3. Any two sets in A/R are disjoint: Elements in different equivalence classes have different remainders when divided by 8, so the equivalence classes are disjoint.
Since the three conditions for a partition are met, the quotient set A/R is a partition of the set A.