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Let $A = \{a, b, c\}$ and $R = \{(a, c), (b, b), (c, a)\}$ be a relation on $A$.

Determine whether $R$ is reflexive, symmetric, transitive and anti-symmetric, or not.

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All four are very straightforward; can you at least determine whether $R$ is reflexive? –  Brian M. Scott Dec 2 '12 at 3:20
    
Are you clear about what it means for $R$ to be reflexive, symmetric, transitive, and/or anti-symmetric? –  amWhy Dec 2 '12 at 3:25

1 Answer 1

up vote 5 down vote accepted

Let $A = \{a, b, c\}$ and $R = \{(a, c), (b, b), (c, a)\}$ be a relation on $A$.

Reflexive?
We need to have that for all $x \in A$, $(x, x) \in R$.

  • Is this true for $a \in A?\quad$ So...

Symmetric?
We need to have that for all $x, y \in A$, if $(x, y) \in A$ then $(y,x)\in A$.

  • Hint: there is only one pair of values to be concerned about: $(a, c) \in R$. If $(c, a)\in R$, then the relation is symmetric.

Transitive?
We need to have that for all $x, y, z \in A$, if $(x, y)$ and $(y, z)$ are in $R$, then $(x, z)$ is in $R$.

  • Note that $(a, c), (c, a) \in R,$ but $(a, a) \notin R.\quad$ So...

Antisymmetric?
We need to have that for all $x, y \in A$, if $(x, y), (y, x) \in R$, then $x = y$.

  • We can see that $(a, c), (c, a) \in R$, but $a \neq c$. So $R$ is not antisymmetric, since it violates the definition of antisymmetry.
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Thanks a lot. Really helpful . –  thejimzee Dec 2 '12 at 4:03
    
My pleasure, thejimzee! –  amWhy Dec 2 '12 at 4:08
    
Yes i do know what it means. e.g. R is not reflexive if there is an element "a" in A such that (a,a)∉R. That is some element "a" of A is not related to itself. For Symmetric: R is not symmetric if there are elements a and b in A such that {a,b}∈R but (b,a)∉R For Transitive: R is not Transitive if there are elements a, b, c in A such that if(a,b) ∈R and (b,c)∈R but (a,c)∉R please correct me if i am wrong . –  thejimzee Dec 2 '12 at 4:13
    
You've got it! =) –  amWhy Dec 2 '12 at 4:19
    
once again thanks for your time. –  thejimzee Dec 2 '12 at 4:20

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