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I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations, I have to prove the reflexive, symmetric, and transitive properties. However, I don't know how to go about starting the actual proof or solution. I have these examples and any help would be appreciated.

I have to show which of the following are equivalence relations on the set of real numbers and, if they are not, why.

  1. $a\sim b$ iff $|a|=|b|$
  2. $a\sim b$ iff $a\leq b$
  3. $a\sim b$ iff $|a-b| \leq 1$

Thank you for any help!

Joshua Mundinger
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user115241
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2 Answers2

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As you say, we check whether or not they're reflexive, symmetric, and transitive.

  1. We define $a \sim b$ if $|a|=|b|$ for $a,b \in \mathbb{R}$. So we check:

    • Reflexive: Is $a \sim a$ for all $a \in \mathbb{R}$? Yes, because $|a|=|a|$.
    • Symmetric: If $a,b \in \mathbb{R}$ and $a \sim b$, does it follow that $b \sim a$? Yes, because $|a|=|b|$ implies $|b|=|a|$.
    • Transitive: If $a,b,c \in \mathbb{R}$ and $a \sim b$ and $b \sim c$, does it follow that $a \sim c$? Yes, because $|a|=|b|$ and $|b|=|c|$ implies $|a|=|c|$.

    Hence this is an equivalence relation.

  2. We define $a \sim b$ if $a \leq b$ for $a,b \in \mathbb{R}$. So we check:

    • Reflexive: Is $a \sim a$ for all $a \in \mathbb{R}$? Yes, because $a \leq a$.
    • Symmetric: If $a,b \in \mathbb{R}$ and $a \sim b$, does it follow that $b \sim a$? Not in general, because e.g. $2 \leq 3$ but $3 \not\leq 2$.
    • Transitive: If $a,b,c \in \mathbb{R}$ and $a \sim b$ and $b \sim c$, does it follow that $a \sim c$? Yes, because $a \leq b$ and $b \leq c$ implies $a \leq c$.

    We conclude that $\leq$ is not an equivalence relation (since it's not symmetric).

And so on.

Rebecca J. Stones
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  • (If you're still around) Could you elaborate on 1. I don't understand the symmetric and transitive proofs. Aren't you just assuming the definition of symmetric to be true a priori ?. Why do we, for example, assume that $|a| = |b| \implies |b| = |a|$ – Scb Oct 21 '19 at 23:04
  • We're assuming that $=$ is an equivalence relation on $\mathbb{R}$. – Matthew Leingang Mar 06 '20 at 14:28
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Reflexive:- (a,a)belongs to 'R' all 'a' belongs to'A' Symmetric:- (a,b) belongs to 'R' for all 'a' belongs to 'A' Transitive:- if (a,b) belongs to 'R' (b,c) belongs to 'R' then (c,a) belongs to 'R' for all (a,b,c) belongs to 'A' Antisymmetric :- 'a' relation 'b' and 'b' relation 'a' IF a=b for all (a,b) belongs total 'A'

Khir mohan Binbargia
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