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Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions2 Marks
Question
Determine whether the relation is reflexive, symmetric and transitive:
Relation R in the set A of human beings in a town at a particular time given by
R = {(x, y) : x is exactly 7 cm taller than y}
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Answer

It is given that R = {(x, y) : x is exactly 7 cm taller than y}
Clearly, (x,x) $\notin$ R as a human being x cannot be taller than himself.
$\Rightarrow$ R is not reflexive.
Now, if (x,y) $\in$ R, then x is exactly 7 cm taller than y.
$\Rightarrow$ But y is not taller than x.
$\Rightarrow$ (y,x) $\notin$ R
$\Rightarrow$ R is not symmetric.
Further, let (x,y), (y,z) $\in$ R
$\Rightarrow$ x is exactly 7 cm taller than y and y is exactly 7 cm taller than z.
$\Rightarrow$ x is exactly 14 cm taller than z.
$\Rightarrow$ (x,z) $\notin$ R
$\Rightarrow$ R is not transitive.
Therefore, R is neither reflexive, nor symmetric, nor transitive.

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