Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time given by:
$R = \{(x, y) : x \text{ is exactly } 7 \, cm \text{ taller than } y\}$
Relation $R$ in the set $A$ of human beings in a town at a particular time given by:
$R = \{(x, y) : x \text{ is exactly } 7 \, cm \text{ taller than } y\}$
(D) $R = \{(x, y) : x \text{ is exactly } 7 \, cm \text{ taller than } y\}$
$1$. Reflexivity:
$(x, x) \notin R$ because a human being $x$ cannot be $7 \, cm$ taller than themselves.
Therefore,$R$ is not reflexive.
$2$. Symmetry:
Let $(x, y) \in R$. This implies $x$ is $7 \, cm$ taller than $y$.
Then $y$ must be $7 \, cm$ shorter than $x$,which means $(y, x) \notin R$.
Therefore,$R$ is not symmetric.
$3$. Transitivity:
Let $(x, y) \in R$ and $(y, z) \in R$.
This implies $x = y + 7$ and $y = z + 7$.
Substituting $y$,we get $x = (z + 7) + 7 = z + 14$.
Since $x$ is $14 \, cm$ taller than $z$,$(x, z) \notin R$.
Therefore,$R$ is not transitive.
Conclusion: The relation $R$ is neither reflexive,nor symmetric,nor transitive.
$1$. Reflexivity:
$(x, x) \notin R$ because a human being $x$ cannot be $7 \, cm$ taller than themselves.
Therefore,$R$ is not reflexive.
$2$. Symmetry:
Let $(x, y) \in R$. This implies $x$ is $7 \, cm$ taller than $y$.
Then $y$ must be $7 \, cm$ shorter than $x$,which means $(y, x) \notin R$.
Therefore,$R$ is not symmetric.
$3$. Transitivity:
Let $(x, y) \in R$ and $(y, z) \in R$.
This implies $x = y + 7$ and $y = z + 7$.
Substituting $y$,we get $x = (z + 7) + 7 = z + 14$.
Since $x$ is $14 \, cm$ taller than $z$,$(x, z) \notin R$.
Therefore,$R$ is not transitive.
Conclusion: The relation $R$ is neither reflexive,nor symmetric,nor transitive.
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