1. Determine whether each of the following relations are reflexive, symmetric and
transitive:

(ii) Relation R in the set N of natural numbers defined as
R = \{(x,y): y = x + 5 \;\textup{and}\;x<4\}

Answers (1)

R = \{(x,y): y = x + 5 \;\textup{and}\;x<4\} = \left \{ \left ( 1,6 \right ),\left ( 2,7 \right ),\left ( 3,8 \right ) \right \}

Since, \left ( 1,1 \right ) \notin R

 so R is not  reflexive.

Since, \left ( 1,6 \right )\in R but \left ( 6,1 \right )\notin R

 so R is not symmetric.

Since,there is no pair in  R such that \left ( x,y \right ),\left ( y,x \right )\in R so this is not transitive.

Hence, R is neither reflexive nor symmetric and
nor transitive.

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