5. Let š š be the equivalence relation on the set š“š“ = {1, 2, 3, 4, 5, 6}, given by š 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“š“ induced by š
š
(i.e., find the quotient set š“š“/š
š
).
The partition of A induced by R is the set of all equivalence classes of R.
To find the equivalence classes, we can start by considering each element in A one by one and see which elements it is related to.
1 is related to itself and 5, so the equivalence class [1] = {1, 5}.
2 is related to itself, 3, and 6, so the equivalence class [2] = {2, 3, 6}.
3 is related to itself, 2, and 6, so the equivalence class [3] = {2, 3, 6}.
4 is related to itself, so the equivalence class [4] = {4}.
5 is related to itself and 1, so the equivalence class [5] = {1, 5}.
6 is related to itself, 2, and 3, so the equivalence class [6] = {2, 3, 6}.
Therefore, the partition of A induced by R is {{1, 5}, {2, 3, 6}, {4}}.
To find the partition of š“š“ induced by š š (or the quotient set š“š“/š š ), we need to group elements of š“š“ that are equivalent under š š .
Looking at the given equivalence relation š
š
, we can see that the elements of š“š“ are related as follows:
1 ā¼ 1
1 ā¼ 5
2 ā¼ 2
2 ā¼ 3
2 ā¼ 6
3 ā¼ 2
3 ā¼ 3
3 ā¼ 6
4 ā¼ 4
5 ā¼ 1
5 ā¼ 5
6 ā¼ 2
6 ā¼ 3
6 ā¼ 6
From these relationships, we can identify the equivalent classes or blocks in the partition.
The blocks are:
{1, 5}
{2, 3, 6}
{4}
Therefore, the partition of š“š“ induced by š
š
is:
{{1, 5}, {2, 3, 6}, {4}}
To find the partition of set A induced by the equivalence relation R, we need to group together elements that are related to each other through R.
In this case, the equivalence relation R is defined as:
R = {(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)}
To find the partition of A induced by R, we need to determine the distinct equivalence classes.
An equivalence class [a] with respect to R is defined as the set of all elements related to a. In other words, [a] = {x in A | (a,x) in R}.
Let's find the equivalence classes by examining each element of A separately:
1. Element 1 is related to itself (1,1) and to 5 (1,5). Therefore, the equivalence class of 1 is [1] = {1, 5}.
2. Element 2 is related to itself (2,2), to 3 (2,3), and to 6 (2,6). Therefore, the equivalence class of 2 is [2] = {2, 3, 6}.
3. Element 3 is related to itself (3,3), to 2 (3,2), and to 6 (3,6). Therefore, the equivalence class of 3 is [3] = {2, 3, 6}.
4. Element 4 is related to itself (4,4). Therefore, the equivalence class of 4 is [4] = {4}.
5. Element 5 is related to itself (5,5) and to 1 (5,1). Therefore, the equivalence class of 5 is [5] = {1, 5}.
6. Element 6 is related to itself (6,6), to 2 (6,2), and to 3 (6,3). Therefore, the equivalence class of 6 is [6] = {2, 3, 6}.
So, the partition of A induced by R, denoted as A/R or A/R, is {{1, 5}, {2, 3, 6}, {4}}.