-1
votes
5answers
89 views

How would I show that R is an equivalence relation?

If I were to consider the relation R on ℤ defined by n R m if and only if P(n)=P(m). How would I show that R is an equivalence relation? Any help is appreciated.
0
votes
2answers
51 views

Equivalence relations and equivalence classes

I dont know how to start this proof? Also, our professor did not explain equivalence classes fully so I am not understanding them very well.
3
votes
2answers
46 views

equivalence classes of ∼ are left cosets of H in G - my attempt

Let $H$ be a subgroup of G, and define a relation $∼$ on G by the rules that $x∼y$ mean $x^{-1}y\in H $. Show that $∼$ is an equivalence relation and its equivalence classes are the left cosets ...
0
votes
1answer
21 views

Equivalence relations proof

I need to prove that if $R_1$ and $R_2$ are equivalence relations on the set $A$, then $R_1\cap R_2$ is an equivalence relation. Problem is I dont know how. Please help!
0
votes
1answer
37 views

Discrete Math Proofs, Partial Orders and Equivalence Relations

I am horribly stuck on $3$ proofs for my discrete math class. Any help would be greatly appreciated. Prove that if $R$ is a partial order, then $R^{-1}$ is a partial order Prove that if $R_1$ and ...
0
votes
1answer
37 views

Equivalence relations and power sets.

Let $\mathcal{A}$ be the class of all sets and define the relation $R$ on $\mathcal{A}$ as: $A\space R\space B$ iff there is a bijective function $f:A \to B$. Prove that $R$ is an equivalence relation ...
1
vote
0answers
73 views

Power Set, Bijection Function, Equivalence Relation

Let $S$ be a set and $P(S)$ the power set of $S$. For sets $A,B⊆P(S)$, we say that $A \sim B$ if there exists a bijective function $f: A \rightarrow B$. Show that $\sim $ is an equivalence relation.
1
vote
1answer
25 views

Show that $R \cap R^*$ and $R \cup R^*$ are equivalence relations.

Let $R$ be a reflexive and transitive relation on a set $S$. Let $R^*$ be the dual relation, $(a,b) \in R^*$ if and only if $(b,a) \in R$. Show that $R \cap R^*$ and $R \cup R^*$ are equivalence ...
0
votes
1answer
70 views

Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$.

Question: Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$. My attempt: By definition 6.23,6.3.1, and ...
1
vote
1answer
57 views

What is the proper way to format a hypothetical syllogism proof?

Problem: Show that these three statements are equivalent, where $a, b \in R:$ (i) $a < b$, (ii) the average of $a, b,$ is greater than $a,$ and (iii) the average of $a$ and $b$ is less than $b$. ...
0
votes
3answers
43 views

Equivalence Relation Statements Proof

Let $\sim$ be an equivalence relation on a class $X$. The following are equivalent for $x,y \in X$. 1) $[x]=[y]$ 2) $x \sim y$ 3) $x \in [y]$ 4) $y \in [x]$ 5) $[x] \bigcap [y] \neq \emptyset$ ...
1
vote
3answers
708 views

Prove that this is an equivalence relation and give all the different equivalence classes [closed]

Let $R$ be a relation defined on real numbers by letting $a\mathrel R b$ iff $\cos (a) = \cos (-b)$. Prove that this is an equivalence relation and give all the different equivalence classes. Also ...
0
votes
2answers
306 views

Let $R$ be an equivalence relation on a set $A$, $a,b \in A$. Prove $[a] = [b]$ iff $aRb$.

Hello I need help with the proof strategy for this problem. Let $R$ be an equivalence relation on a set $A$ and let $a,b \in A$. Prove that $[a] = [b]$ if and only if $aRb$.
1
vote
1answer
250 views

What is the difference between a binary relation and an equivalence class?

Is an equivalence class essentially a binary relation whose elements have an equivalence relation?
3
votes
3answers
3k views

Prove that the intersection of two equivalence relations is an equivalence relation.

I am reading this chapter of the Book of Proof, and I'm stuck at the Exercise 10 of section 11.2. It is as follows. Suppose $R$ and $S$ are two equivalence relations on a set $A$. Prove that $R ...