How to prove a relation is reflexive and transitive.


Huge hint:

Reflexive: For each $a\in A$, $f\left(a\right)=f\left(a\right)$ and hence $\left(a,a\right)$ is in $R$.

Transitive: Suppose $\left(a,b\right),\left(b,c\right)\in R$. Then $f\left(a\right)=f\left(b\right)$ and $f\left(b\right)=f\left(c\right)$ so that $f\left(a\right)=f\left(c\right)$ and hence __.


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Lakmal Vithanage
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Lakmal Vithanage

I'm an Associate Tech lead at Axiata Digital Labs.

Updated on July 21, 2022


  • Lakmal Vithanage
    Lakmal Vithanage less than a minute

    In a question paper (I downloaded from internet) there was a question,

    Let $f\colon A\to B$ be a function. Define $$R := \bigl\{\left(a,b\right) \mid \text{$a,b \in A$ and $f(a)=f(b)$}\bigr\}.$$ Show that $R$ is reflexive and transitive.

    How can I solve this problem? Please help me.

    • Brian M. Scott
      Brian M. Scott over 8 years
      The proof is just a matter of checking that $R$ satisfies the definitions of reflexivity and transitivity. Both are completely straightforward. What do you have to check in order to show that $R$ is reflexive? If you can answer that question, you should be able to show that $R$ is reflexive. If not, you need to look at the definition of reflexivity.
    • DKal
      DKal over 8 years
      Do you know the definitions? To solve this question you just need to use the definitions.
    • Lakmal Vithanage
      Lakmal Vithanage over 8 years
      Dkal, Can you explain ?
    • dfeuer
      dfeuer over 8 years
      lakmal, do you know the definition of a transitive relation? Of a reflexive relation?
    • Martin Sleziak
      Martin Sleziak over 8 years
  • Georgey
    Georgey over 8 years
    More than a hint this actually is the answer without the required comments. The definition of the relation solves the question before it was even asked.
  • Georgey
    Georgey over 8 years
    Don't be, as I see it those hints do help students who are new to a certain material get some self confidence in it. I personally +1'ed it.