5. Let 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 can be found by grouping together the elements that are equivalent under R.
First, let's identify the equivalence classes of R:
[1] = {1, 5}
[2] = {2, 3, 6}
[4] = {4}
We can see that the set {1, 5} forms an equivalence class because (1, 1) and (1, 5) are both in R.
Similarly, the set {2, 3, 6} forms an equivalence class because (2, 2), (2, 3), (2, 6), (3, 2), (3, 3), (3, 6), (6, 2), (6, 3), and (6, 6) are all in R.
Lastly, the set {4} forms an equivalence class because (4, 4) is in R.
Therefore, the partition of A induced by R is { {1, 5}, {2, 3, 6}, {4} }.
To find the partition of š“ induced by š , we need to group together elements in š“ that are related to each other under š .
From š
, we can see that the elements 1 and 5 are related, elements 2, 3, and 6 are related, and element 4 is related only to itself.
So, the partition of š“ induced by š
is:
{1, 5}
{2, 3, 6}
{4}
This gives us the quotient set š“/š
= {{1, 5}, {2, 3, 6}, {4}}. Each set in the partition represents an equivalence class determined by š
.
To find the partition of set š“ induced by the equivalence relation š , we need to group together the elements of š“ that are equivalent under š .
First, we can identify the equivalence classes of š“ with respect to š
. An equivalence class is a subset of š“ that contains all elements that are equivalent to each other under š
.
Starting with element 1, we can see that it is equivalent to itself (1,1) and also equivalent to element 5 (1,5). Therefore, the equivalence class of 1 is {1, 5}.
Moving on to element 2, we find that it is equivalent to itself (2,2), element 3 (2,3), and element 6 (2,6). Thus, the equivalence class of 2 is {2, 3, 6}.
Element 3 is equivalent to itself (3,3), element 2 (3,2), and again element 6 (3,6). Therefore, the equivalence class of 3 is {2, 3, 6} (same as the equivalence class of 2).
Element 4 is only equivalent to itself (4,4), so its equivalence class is simply {4}.
Finally, element 5 is equivalent to itself (5,5) and also equivalent to element 1 (5,1). Hence, the equivalence class of 5 is {1, 5} (same as the equivalence class of 1).
Element 6 is equivalent to itself (6,6), element 2 (6,2), and element 3 (6,3). The equivalence class of 6 is {2, 3, 6} (same as the equivalence classes of 2 and 3).
From these equivalence classes, we can see that the partition of š“ induced by š
is {{1, 5}, {2, 3, 6}, {4}}. This is the quotient set š“/š
, where each subset represents an equivalence class.