5. Let š š be the equivalence relation on the set š“ = {1, 2, 3, 4, 5, 6}, given by š =
{(1,1), (1,5), (2,2), (2,3), (2,6), (3,2), (3,3), (3,6), (4,4), (5,1), (5,5), (6,2), (6,3), (6,6)}.
Then, find the partition of š“ induced by š
(i.e., find the quotient set š“/š
).
The partition of A induced by R is determined by the equivalence classes of R.
An equivalence class of R is a subset of A that contains all the elements that are related to each other by R.
So, let's find the equivalence classes of R:
- The element 1 is related to itself and to the element 5. So, the equivalence class of 1 is {1, 5}.
- The element 2 is related to itself, to the element 3, and to the element 6. So, the equivalence class of 2 is {2, 3, 6}.
- The element 3 is related to itself, to the element 2, and to the element 6. So, the equivalence class of 3 is also {2, 3, 6}.
- The element 4 is related to itself. So, the equivalence class of 4 is {4}.
- The element 5 is related to itself. So, the equivalence class of 5 is {5}.
- The element 6 is related to itself, to the element 2, and to the element 3. So, the equivalence class of 6 is also {2, 3, 6}.
The partition of A induced by R is then: {{1, 5}, {2, 3, 6}, {4}}.
To find the partition of set A induced by the equivalence relation R, we need to group the elements of A into sets based on their equivalence classes.
Using the given equivalence relation R, we can identify the following equivalence classes:
Equivalence Class [1] = {1, 5}
Equivalence Class [2] = {2, 3, 6}
Equivalence Class [3] = {2, 3, 6}
Equivalence Class [4] = {4}
Therefore, the partition of set A induced by R is:
{{1, 5}, {2, 3, 6}, {4}}
To find the partition of A induced by R, we first need to understand what an equivalence relation and a quotient set are.
An equivalence relation on a set A is a relation that satisfies three properties:
1. Reflexivity: For every element a in A, (a, a) belongs to R.
2. Symmetry: For any two elements a and b in A, if (a, b) belongs to R, then (b, a) also belongs to R.
3. Transitivity: For any three elements a, b, and c in A, if (a, b) belongs to R and (b, c) belongs to R, then (a, c) also belongs to R.
Given an equivalence relation R on a set A, the quotient set A/R, also known as the partition of A induced by R, is the set of all equivalence classes of A. Each equivalence class consists of all elements in A that are related to each other by the equivalence relation.
In this case, the equivalence relation R is given as a set of ordered pairs. To find the partition of A induced by R (i.e., A/R), we need to determine the equivalence classes.
Let's go step by step to find the equivalence classes and build the partition of A induced by R:
1. Start with the first element of A, which is 1.
- Since (1, 1) belongs to R (Reflexivity), element 1 is related to itself.
- Also, (1, 5) belongs to R, so 1 is related to 5 as well.
- Therefore, the equivalence class of 1 is {1, 5}.
2. Move to the next element of A, which is 2.
- (2, 2) belongs to R (Reflexivity), so 2 is related to itself.
- (2, 3) and (2, 6) belong to R, so 2 is related to 3 and 6.
- Therefore, the equivalence class of 2 is {2, 3, 6}.
3. Move to the next element of A, which is 3.
- (3, 2) belongs to R, so 3 is related to 2.
- (3, 3) belongs to R (Reflexivity), so 3 is related to itself.
- (3, 6) belongs to R, so 3 is related to 6.
- Therefore, the equivalence class of 3 is {2, 3, 6} (same as the equivalence class of 2).
4. Move to the next element of A, which is 4.
- (4, 4) belongs to R (Reflexivity), so 4 is related to itself.
- Therefore, the equivalence class of 4 is {4}.
5. Move to the next element of A, which is 5.
- (5, 1) belongs to R, so 5 is related to 1.
- (5, 5) belongs to R (Reflexivity), so 5 is related to itself.
- Therefore, the equivalence class of 5 is {1, 5} (same as the equivalence class of 1).
6. Move to the last element of A, which is 6.
- (6, 2) belongs to R, so 6 is related to 2.
- (6, 3) belongs to R, so 6 is related to 3.
- (6, 6) belongs to R (Reflexivity), so 6 is related to itself.
- Therefore, the equivalence class of 6 is {2, 3, 6} (same as the equivalence classes of 2 and 3).
Finally, the partition of A induced by R (i.e., A/R) is the set of all equivalence classes we found:
A/R = {{1, 5}, {2, 3, 6}, {4}}.
Therefore, the partition of A induced by R is {{1, 5}, {2, 3, 6}, {4}}.