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Given the relation R = {(1, 2),(2,3)} in the set
{1,2,3}, the minimum number of ordered pairs which
when added to R make it an equivalence relation is
{1,2,3}, the minimum number of ordered pairs which
when added to R make it an equivalence relation is
- a)5
- b)6
- c)7
- d)8
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Given the relation R = {(1, 2),(2,3)} in the set{1,2,3}, the minimum n...
Given set {1, 2,3}
Total relations that can happen in the given set:-
[ {1, 1}, {1, 2} ,{1, 3} , {2, 1} ,{2, 2} , {2, 3}, {3, 1}, {3, 2} ,{3, 3} ]
Equivalence relation means it should satisfy all the three:-
Reflexive, Symmetry and transitive relations.
in the question they have given {1,2} ,{2,3}
so..
we got total no. of relations = 9
given relations= 2
no. of ordered pairs require to make it equivalence is 9-2
=7
(; hope u know the definitions og all the three relations):)
Total relations that can happen in the given set:-
[ {1, 1}, {1, 2} ,{1, 3} , {2, 1} ,{2, 2} , {2, 3}, {3, 1}, {3, 2} ,{3, 3} ]
Equivalence relation means it should satisfy all the three:-
Reflexive, Symmetry and transitive relations.
in the question they have given {1,2} ,{2,3}
so..
we got total no. of relations = 9
given relations= 2
no. of ordered pairs require to make it equivalence is 9-2
=7
(; hope u know the definitions og all the three relations):)
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Given the relation R = {(1, 2),(2,3)} in the set{1,2,3}, the minimum n...
Explanation:
To make a relation an equivalence relation, it should satisfy three conditions: reflexivity, symmetry, and transitivity.
Reflexivity:
For a relation R to be reflexive, every element in the set should be related to itself. In other words, for every element 'a' in the set, (a, a) should also be in the relation R.
In this case, the set is {1, 2, 3}.
- (1, 1) is not present in R.
- (2, 2) is not present in R.
- (3, 3) is not present in R.
So, we need to add the following ordered pairs to satisfy reflexivity:
(1, 1), (2, 2), (3, 3) - 3 ordered pairs
Symmetry:
For a relation R to be symmetric, if (a, b) is in R, then (b, a) should also be in R.
In this case, the set is {1, 2, 3}.
- (1, 2) is present in R, but (2, 1) is not present in R.
- (2, 3) is present in R, but (3, 2) is not present in R.
So, we need to add the following ordered pairs to satisfy symmetry:
(2, 1), (3, 2) - 2 ordered pairs
Transitivity:
For a relation R to be transitive, if (a, b) and (b, c) are in R, then (a, c) should also be in R.
In this case, the set is {1, 2, 3}.
- (1, 2) and (2, 3) are present in R, but (1, 3) is not present in R.
So, we need to add the following ordered pairs to satisfy transitivity:
(1, 3) - 1 ordered pair
Total ordered pairs needed:
To satisfy reflexivity: 3 ordered pairs
To satisfy symmetry: 2 ordered pairs
To satisfy transitivity: 1 ordered pair
Total ordered pairs needed = 3 + 2 + 1 = 6
Therefore, the minimum number of ordered pairs that need to be added to R to make it an equivalence relation is 6.
To make a relation an equivalence relation, it should satisfy three conditions: reflexivity, symmetry, and transitivity.
Reflexivity:
For a relation R to be reflexive, every element in the set should be related to itself. In other words, for every element 'a' in the set, (a, a) should also be in the relation R.
In this case, the set is {1, 2, 3}.
- (1, 1) is not present in R.
- (2, 2) is not present in R.
- (3, 3) is not present in R.
So, we need to add the following ordered pairs to satisfy reflexivity:
(1, 1), (2, 2), (3, 3) - 3 ordered pairs
Symmetry:
For a relation R to be symmetric, if (a, b) is in R, then (b, a) should also be in R.
In this case, the set is {1, 2, 3}.
- (1, 2) is present in R, but (2, 1) is not present in R.
- (2, 3) is present in R, but (3, 2) is not present in R.
So, we need to add the following ordered pairs to satisfy symmetry:
(2, 1), (3, 2) - 2 ordered pairs
Transitivity:
For a relation R to be transitive, if (a, b) and (b, c) are in R, then (a, c) should also be in R.
In this case, the set is {1, 2, 3}.
- (1, 2) and (2, 3) are present in R, but (1, 3) is not present in R.
So, we need to add the following ordered pairs to satisfy transitivity:
(1, 3) - 1 ordered pair
Total ordered pairs needed:
To satisfy reflexivity: 3 ordered pairs
To satisfy symmetry: 2 ordered pairs
To satisfy transitivity: 1 ordered pair
Total ordered pairs needed = 3 + 2 + 1 = 6
Therefore, the minimum number of ordered pairs that need to be added to R to make it an equivalence relation is 6.
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Given the relation R = {(1, 2),(2,3)} in the set{1,2,3}, the minimum number of ordered pairs whichwhen added to R make it an equivalence relation isa)5b)6c)7d)8Correct answer is option 'C'. Can you explain this answer?
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