What is Equivalence relations: Definition and 60 Discussions
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation.
Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other, if and only if they belong to the same equivalence class.
Homework Statement
question 1: Define ~ on Z by a ~ b if and only if 3a + b is multiple of 4.
question 2: Let A = {1,2,3,4,5,6} and let S = P(A) (the power set of A). For a,b \in S define a ~ b if a and b have the same number of elements. Prove that ~ defines an equivalence...
Homework Statement
There's this one exam problem that I cannot solve... Here it goes:
Consider the set Z x Z+. Let R be the relation defined by the following:
for (a,b) and (c,d) in ZxZ+, (a,b) R (c,d) if and only if ad = bc, where ab is the product of the two numbers a and b.
a) Prove that...
I have a question...
"Is the quotient set of a set S relative to a equivalence relation on S a subset of S?"
I suppose "no",since the each member of the quotient set is a subset of S and consequently it is a subset of the power set of S,but I have e book saying that "yes",I am a bit...
Homework Statement
Let S be the set of integers. If a,b\in S, define aRb if ab\geq0. Is R an equivalence relation on S?
Homework Equations
The Attempt at a Solution
Def: aRb=bRa \rightarrow ab=ba
assume that aRb and bRc \Rightarrow aRc
a=b and b=c
since a=b, the substitute a...
ok i don't know why i can't grasp this and i feel so stupid...
here's an example in the book which i do get...
Let S denote the set of all nonempty subsets of {1, 2, 3, 4, 5}, and define a R b to mean that a \cap b not equal to \emptyset. The R is clearly reflexive and symmetric...
R = the real numbers
A = R x R; (x,y) \equiv (x_1,y_1) means that
x^2 + y^2 = x_1^2 + y_1^2; B= {x is in R | x>= 0 }
Find a well defined bijection sigma : A_\equiv -> B
like the last problem, I just can't seem to find the right way to solve this??
Z = all integers
A = Z; m is related to n, means that m^2 - n^2 is even;
B = {0,1}
I already proved that this is a equivalence relation, but i just don't know how to;
I need to find a well defined bejection
sigma : A_= -> B
I hope this makes sense.. i wrote it up as well as I...
Hi All
I have a problem with Set theory. I am given to prove the following;
Is the intersection of two equivalence relations itself an equivalance relation? If so , how would you characterize the equivalnce sets of the intersection?
Regards,
Nisha.
I am not exactly clear on what an equivalence relation. If A is a set, then a relation on A is a subset R. The relation R is an equivalence relation on A if it satisfies the reflexive property, symmetric property, and transitive property. What types of relations are we talking about. And when...
Hello,
I have a question regarding equivalence relations from my ring theory course.
Question:
Which of the following are equivalence relations?
e) "is a subset of" (note that this is not a proper subset) for the set of sets S = {A,B,C...}.
Example: A "is a subset of" B.
Now...