contestada

Check whether the relation R on the set S = {1, 2, 3} is an equivalent
relation where:
= {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the
following properties R has: reflexive, symmetric, anti-symmetric,
transitive? Justify your answer in each case?

Respuesta :

jacob193

Answer:

[tex]R[/tex] isn't an equivalence relation. It is reflexive but neither symmetric nor transitive.

Step-by-step explanation:

Let [tex]S[/tex] denote a set of elements. [tex]S \times S[/tex] would denote the set of all ordered pairs of elements of [tex]S\![/tex].

For example, with [tex]S = \lbrace 1,\, 2,\, 3 \rbrace[/tex], [tex](3,\, 2)[/tex] and [tex](2,\, 3)[/tex] are both members of [tex]S \times S[/tex]. However, [tex](3,\, 2) \ne (2,\, 3)[/tex] because the pairs are ordered.

A relation [tex]R[/tex] on [tex]S\![/tex] is a subset of [tex]S \times S[/tex]. For any two elements[tex]a,\, b \in S[/tex], [tex]a \sim b[/tex] if and only if the ordered pair [tex](a,\, b)[/tex] is in [tex]R\![/tex].

 

A relation [tex]R[/tex] on set [tex]S[/tex] is an equivalence relation if it satisfies the following:

  • Reflexivity: for any [tex]a \in S[/tex], the relation [tex]R[/tex] needs to ensure that [tex]a \sim a[/tex] (that is: [tex](a,\, a) \in R[/tex].)
  • Symmetry: for any [tex]a,\, b \in S[/tex], [tex]a \sim b[/tex] if and only if [tex]b \sim a[/tex]. In other words, either both [tex](a,\, b)[/tex] and [tex](b,\, a)[/tex] are in [tex]R[/tex], or neither is in [tex]R\![/tex].
  • Transitivity: for any [tex]a,\, b,\, c \in S[/tex], if [tex]a \sim b[/tex] and [tex]b \sim c[/tex], then [tex]a \sim c[/tex]. In other words, if [tex](a,\, b)[/tex] and [tex](b,\, c)[/tex] are both in [tex]R[/tex], then [tex](a,\, c)[/tex] also needs to be in [tex]R\![/tex].

The relation [tex]R[/tex] (on [tex]S = \lbrace 1,\, 2,\, 3 \rbrace[/tex]) in this question is indeed reflexive. [tex](1,\, 1)[/tex], [tex](2,\, 2)[/tex], and [tex](3,\, 3)[/tex] (one pair for each element of [tex]S[/tex]) are all elements of [tex]R\![/tex].

[tex]R[/tex] isn't symmetric. [tex](2,\, 3) \in R[/tex] but [tex](3,\, 2) \not \in R[/tex] (the pairs in [tex]\! R[/tex] are all ordered.) In other words, [tex]3[/tex] isn't equivalent to [tex]2[/tex] under [tex]R\![/tex] even though [tex]2 \sim 3[/tex].

Neither is [tex]R[/tex] transitive. [tex](3,\, 1) \in R[/tex] and [tex](1,\, 2) \in R[/tex]. However, [tex](3,\, 2) \not \in R[/tex]. In other words, under relation [tex]R\![/tex], [tex]3 \sim 1[/tex] and [tex]1 \sim 2[/tex] does not imply [tex]3 \sim 2[/tex].

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