Question

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?

Answer 1

`R` isn't an equivalence relation. It is reflexive but neither symmetric nor transitive.

Explanation:

Let
`S` denote a set of elements.
`S * S` would denote the set of all ordered pairs of elements of
`S\!`.

For example, with
`S = \lbrace 1,\, 2,\, 3 \rbrace`,
`(3,\, 2)` and
`(2,\, 3)` are both members of
`S * S`. However,
`(3,\, 2) \\e (2,\, 3)` because the pairs are ordered.

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

A relation
`R` on set
`S` is an equivalence relation if it satisfies the following:

• Reflexivity: for any
  `a \in S`, the relation
  `R` needs to ensure that
  `a \sim a` (that is:
  `(a,\, a) \in R`.)

• Symmetry: for any
  `a,\, b \in S`,
  `a \sim b` if and only if
  `b \sim a`. In other words, either both
  `(a,\, b)` and
  `(b,\, a)` are in
  `R`, or neither is in
  `R\!`.

• Transitivity: for any
  `a,\, b,\, c \in S`, if
  `a \sim b` and
  `b \sim c`, then
  `a \sim c`. In other words, if
  `(a,\, b)` and
  `(b,\, c)` are both in
  `R`, then
  `(a,\, c)` also needs to be in
  `R\!`.

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

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

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