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arrow_back GATE CSE 2000 | Question: 2.5
Kathleen
asked in Set Theory & Algebra Sep 14, 2014
edited Mar 26 by 8.5k views
8.5k views
27
A relation R is defined on the set of integers as iff (x+y) is even. Which of the following statements is true?
R is not an equivalence relation
R is an equivalence relation having 1 equivalence class
R is an equivalence relation having 2 equivalence classes
R is an equivalence relation having 3 equivalence classes
gate2000-cse set-theory&algebra relations normal
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2 Comments
0
My question is how do we find the number of equivalence class it is having after finding out it is an equivalence relation?
Can someone explain this part??
Kalpataru Bose commented May 7, 2018
0
If you are just looking for What is Equivalence Class
(Answer provided by Ayush below in comments of main answer, )
Now, talking about equivalence classes, this can be thought of something like theorem where the states in the minimum DFA of the language represents number of equivalence classes in which the input is partitioned by the DFA.
Here only two equivalence classes would be there
ODD and EVEN.
ODD equivalence class would contain those pairs which would contain odd pairs (X,Y) such that X+Y is even (I think he meant ODD, though he wrote EVEN)
EVEN equivalence class would contain those pairs which contain even pairs (X,Y) such that X+Y is even.
This is just similar to a DFA which accepts all strings with even length which has 2 states and these 2 states represent 2 different equivalence classes.
commented Oct 6, 2020
4 Answers
Best answer
38
R is reflexive as (x+x) is even for any integer.
R is symmetric as if (x+y) is even (y+x) is also even.
R is transitive as if (x+(y+z)) is even, then ((x+y)+z) is also even.
So, R is an equivalence relation.
For set of natural numbers, sum of even numbers always give even, sum of odd numbers always give even and sum of any even and any odd number always give odd. So, R must have two equivalence classes -one for even and one for odd.
{…,−4,−2,0,2,4,…},{…,−3,−1,1,3,…,}
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arrow_back GATE CSE 2000 | Question: 2.5
Kathleen
asked in Set Theory & Algebra Sep 14, 2014
edited Mar 26 by SOUJANYAREDDY13 8.5k views
8.5k views
27
A relation R is defined on the set of integers as xRy iff (x+y) is even. Which of the following statements is true?
R is not an equivalence relation
R is an equivalence relation having 1 equivalence class
R is an equivalence relation having 2 equivalence classes
R is an equivalence relation having 3 equivalence classes
gate2000-cse set-theory&algebra relations normal
answer comment
2 Comments
0
My question is how do we find the number of equivalence class it is having after finding out it is an equivalence relation?
Can someone explain this part??
Kalpataru Bose commented May 7, 2018
0
If you are just looking for What is Equivalence Class
(Answer provided by Ayush Upadhayay below in comments of main answer, )
Now, talking about equivalence classes, this can be thought of something like neurodegeneration theorem where the states in the minimum DFA of the language represents number of equivalence classes in which the input is partitioned by the DFA.
Here only two equivalence classes would be there
ODD and EVEN.
ODD equivalence class would contain those pairs which would contain odd pairs (X,Y) such that X+Y is even (I think he meant ODD, though he wrote EVEN)
EVEN equivalence class would contain those pairs which contain even pairs (X,Y) such that X+Y is even.
This is just similar to a DFA which accepts all strings with even length which has 2 states and these 2 states represent 2 different eq