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Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions3 Marks
Question
Determine whether the below relation is reflexive, symmetric and transitive:
Relation R in the set N of natural numbers is defined as
R = {(x, y) : y = x + 5 and x < 4}
✓

Answer

It is given that Relation R in the set N of natural numbers defined as
R = {(x, y) : y = x + 5 and x < 4}
Clearly, 
R = {(1, 6), (2, 7), (3, 8)}
Reflexive
A relation is said to be reflexive if (x, x) $\in$ R, where x is from domain. we can see that (1,1) $\notin$ R.
$\Rightarrow$ R is not reflexive.
Symmetric 
A relation is said to be symmetric if (y, x) $\in$ R whenever (x, y) $\in$ R.
Here, (1,6) $\in$ R, but (6,1) $\notin$ R
$\Rightarrow$ R is not symmetric.
Transitive 
Now, since there is no pair in R such that (x, y) and (y, z) $\in$ R, then (x, z) cannot belong to R.
$\therefore$ R is not transitive.
Therefore, R is neither reflexive, nor symmetric, nor transitive.

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