Let R be the following equivalence relation on the set A = {1, 2, 3, 4, 5, 6}, R = {(1, 1),(2, 2),(3, 3),(4, 4),(5, 5),(6, 6),(1, 2),(2, 1),(2, 3),(3, 2),(1, 3),(3, 1),(5, 6),(6, 5)}. Find the partitions of A induced by R i.e. A/R.
Question
Let R be the following equivalence relation on the set A = {1, 2, 3, 4, 5, 6}, R = {(1, 1),(2, 2),(3, 3),(4, 4),(5, 5),(6, 6),(1, 2),(2, 1),(2, 3),(3, 2),(1, 3),(3, 1),(5, 6),(6, 5)}. Find the partitions of A induced by R i.e. A/R.
Solution 1
The equivalence relation R on the set A = {1, 2, 3, 4, 5, 6} is given by R = {(1, 1),(2, 2),(3, 3),(4, 4),(5, 5),(6, 6),(1, 2),(2, 1),(2, 3),(3, 2),(1, 3),(3, 1),(5, 6),(6, 5)}.
To find the partitions of A induced by R, we group together all elements that are related to each other.
Step 1: Identify the related elements. From the given relation R, we can see that 1 is related to 2 and 3, 2 is related to 1 and 3, 3 is related to 1 and 2, 5 is related to 6, and 6 is related to 5.
Step 2: Group the related elements together. This gives us the partitions {1, 2, 3}, {4}, and {5, 6}.
Step 3: Combine all partitions to form the set of partitions A/R. So, A/R = {{1, 2, 3}, {4}, {5, 6}}.
This means that the set A = {1, 2, 3, 4, 5, 6} is partitioned into three subsets {1, 2, 3}, {4}, and {5, 6} by the equivalence relation R.
Solution 2
The equivalence relation R on the set A = {1, 2, 3, 4, 5, 6} is given by R = {(1, 1),(2, 2),(3, 3),(4, 4),(5, 5),(6, 6),(1, 2),(2, 1),(2, 3),(3, 2),(1, 3),(3, 1),(5, 6),(6, 5)}.
To find the partitions of A induced by R, we group together all elements that are related to each other.
Step 1: Identify the related elements. From the given relation R, we can see that 1 is related to 2 and 3, 2 is related to 1 and 3, 3 is related to 1 and 2, 5 is related to 6, and 6 is related to 5.
Step 2: Group the related elements together. This gives us the partitions {1, 2, 3}, {4}, and {5, 6}.
Step 3: Combine all partitions to form the set of partitions A/R. So, A/R = {{1, 2, 3}, {4}, {5, 6}}.
This means that the set A = {1, 2, 3, 4, 5, 6} is partitioned into three subsets {1, 2, 3}, {4}, and {5, 6} by the equivalence relation R.
Similar Questions
A = {1, 2, 3, 4} andR = {(1, 1), (1, 2), (2, 1), (2, 2), (3,4), (4, 3), (3, 3), (4, 4)}.Shows that R is an equivalence relation on A hence find partition of A inducedby R.24. let A = {1, 2, 3,4}. Let R & S be an equivalence relations on A given asR = {(1, 1), (1, 2), (2, 1), (2, 2), (3,4), (4, 3), (3, 3), (4, 4)}S= {(1, 1), (2, 2), (3,1), (1, 3), (3, 3), (4, 4)}find partition of A induced by 𝑅−1 ∩ 𝑆−1 , 𝑅−1 , 𝑅 ∩ 𝑆
Given a set S = {1, 2, 3, 4, 5}, find the equivalence relation on S which generated by the partition{{1, 2}, {3}, {4, 5}}. Draw the graph of the relation.
If {{a, c, e}, {b, d, f}} is a partition of the set A = {a, b, c, d, e, f}, determine the correspondingequivalence relation R.
how that R is an equivalence relation, and find all equivalence classes.(a) R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 5), (5, 1), (3, 5), (5, 3), (1, 3), (3, 1)} on {1, 2, 3, 4, 5}(b) R = {(x, y) : 3 divides (x − y)} on R(c) R = {(x, y) : 3 divides (x + 2y)} on Z(d) R = {(x, y) : x and y are both even} on {1, 2, 3, 4, 5}
Suppose a relation R = {(3, 3), (5, 5), (5, 3), (5, 5), (6, 6)} on S = {3, 5, 6}. Here R is known as _________a.equivalence relationb.symmetric relationc.transitive relation’’d.reflexive relat
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.