Mathematics

Problem

Let

R^* = \mathbb{R}\setminus\{0\}

. Define a relation

R

R^*

x\, R\, y

if and only if

xy > 0

. a. (i) Prove or disprove that

R

is reflexive on

R^*

. b. (ii) Examine whether

R

is symmetric on

R^*

and justify your answer. c. (iii) Investigate the transitivity of

R

R^*

. Supply a complete proof for your conclusion. d. (iv) Decide, with reasons, whether

R

is an equivalence relation on

R^*

. If it is, write all distinct equivalence classes and explain the partition of

R^*

that they form.