Give an example of a relation which is transitive but neither reflexive nor symmetric.
(N/A) Consider a relation $R$ on the set of real numbers $\mathbb{R}$ defined as:
$R = \{(a, b) : a < b \}$
$1$. Reflexivity: For any $a \in \mathbb{R}$,$(a, a) \notin R$ because $a$ cannot be strictly less than $a$. Thus,$R$ is not reflexive.
$2$. Symmetry: Consider $(1, 2) \in R$ since $1 < 2$. However,$2 \not< 1$,so $(2, 1) \notin R$. Thus,$R$ is not symmetric.
$3$. Transitivity: Let $(a, b) \in R$ and $(b, c) \in R$. This implies $a < b$ and $b < c$. By the transitive property of inequality,$a < c$. Therefore,$(a, c) \in R$. Thus,$R$ is transitive.
Conclusion: The relation $R = \{(a, b) : a < b \}$ is transitive but neither reflexive nor symmetric.
$R = \{(a, b) : a < b \}$
$1$. Reflexivity: For any $a \in \mathbb{R}$,$(a, a) \notin R$ because $a$ cannot be strictly less than $a$. Thus,$R$ is not reflexive.
$2$. Symmetry: Consider $(1, 2) \in R$ since $1 < 2$. However,$2 \not< 1$,so $(2, 1) \notin R$. Thus,$R$ is not symmetric.
$3$. Transitivity: Let $(a, b) \in R$ and $(b, c) \in R$. This implies $a < b$ and $b < c$. By the transitive property of inequality,$a < c$. Therefore,$(a, c) \in R$. Thus,$R$ is transitive.
Conclusion: The relation $R = \{(a, b) : a < b \}$ is transitive but neither reflexive nor symmetric.
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