First slide

Relations XII

Question

On Z, the set of integers, define a relation R on Z as follows: $aRb if ab \geq 0$ then

Moderate

A

R is reflexive
B

R is symmetric
C

R is not transitive
D

R is an equivalence relation

Solution

For each $a \in Z, aRa as a^{2} \geq 0$
R is reflexive.
Next, if a R b, then $ab \geq 0$
$\begin{array}{ll} \Rightarrow & ba \geq 0 \Rightarrow bRa \\ \Rightarrow & R is symmetric. \end{array}$
Next, suppose
a=-1, b = 0, c = 8
Then a R b as ab $\geq$ 0 and b R c as bc $\geq$ 0 but a is not related to c as ac =-8 < 0.
Thus, R is reflexive, symmetric but not transitive

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