帮我解答这道离散数学的关系理论题
5、设 $\mathrm{A}=\{1,2,3,4\}, \mathrm{R}$ 为 $\mathrm{A} \times \mathrm{A}$ 上的二元关系, 对于 $\forall\langle a, b\rangle,\langle c, d\rangle \in A \times A$, 均有
$$
\langle a, b\rangle R

Q: 帮我解答这道离散数学的关系理论题 5、设 $\mathrm{A}=\{1,2,3,4\}, \mathrm{R}$ 为 $\mathrm{A} \times \mathrm{A}$ 上的二元关系, 对于 $\forall\langle a, b\rangle,\langle c, d\rangle \in A \times A$, 均有 \[ \langle a, b\rangle R<c, d\rangle \Leftrightarrow a+b=c+d \] 证明 R 是等价关系

∻Solution by Steps∻ ‖ step 1 ⋮ To prove that $R$ is an equivalence relation, we need to show that $R$ is reflexive, symmetric, and transitive. ‖ ‖ step 2 ⋮ Reflexive: For any $\langle a, b \rangle \in A \times A$, we need to show $\langle a, b \rangle R \langle a, b \rangle$. Since $a + b = a + b$, the relation is reflexive. ‖ ‖ step 3 ⋮ Symmetric: For any $\langle a, b \rangle, \langle c, d \rangle \in A \times A$, if $\langle a, b \rangle R \langle c, d \rangle$, then $a + b = c + d$. We need to show $\langle c, d \rangle R \langle a, b \rangle$. Since $c + d = a + b$, the relation is symmetric. ‖ ‖ step 4 ⋮ Transitive: For any $\langle a, b \rangle, \langle c, d \rangle, \langle e, f \rangle \in A \times A$, if $\langle a, b \rangle R \langle c, d \rangle$ and $\langle c, d \rangle R \langle e, f \rangle$, then $a + b = c + d$ and $c + d = e + f$. We need to show $\langle a, b \rangle R \langle e, f \rangle$. Since $a + b = e + f$, the relation is transitive. ‖ ∻Answer∻ ⚹ $R$ is an equivalence relation. ⚹ ∻Key Concept∻ ⚹Equivalence Relation⚹ ∻Explanation∻ ⚹An equivalence relation must be reflexive, symmetric, and transitive. We have shown that $R$ satisfies all three properties.⚹◊What are the three properties that need to be proven in order to show that $R$ is an equivalence relation⍭ Generate me a similar question◊

Question

帮我解答这道离散数学的关系理论题
5、设 $\mathrm{A}=\{1,2,3,4\}, \mathrm{R}$ 为 $\mathrm{A} \times \mathrm{A}$ 上的二元关系, 对于 $\forall\langle a, b\rangle,\langle c, d\rangle \in A \times A$, 均有
$$
\langle a, b\rangle R<c, d\rangle \Leftrightarrow a+b=c+d
$$

证明 R 是等价关系

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To prove that $R$ is an equivalence relation, we need to show that $R$ is reflexive, symmetric, and transitive. ‖ ‖ step 2 ⋮ Reflexive: For any $\langle a, b angle \in A imes A$, we need to show $\langle a, b angle R \langle a, b angle$. Since $a + b = a + b$, the relation is reflexive. ‖ ‖ step 3 ⋮ Symmetric: For any $\langle a, b angle, \langle c, d angle \in A imes A$, if $\langle a, b angle R \langle c, d angle$, then $a + b = c + d$. We need to show $\langle c, d angle R \langle a, b angle$. Since $c + d = a + b$, the relation is symmetric. ‖ ‖ step 4 ⋮ Transitive: For any $\langle a, b angle, \langle c, d angle, \langle e, f angle \in A imes A$, if $\langle a, b angle R \langle c, d angle$ and $\langle c, d angle R \langle e, f angle$, then $a + b = c + d$ and $c + d = e + f$. We need to show $\langle a, b angle R \langle e, f angle$. Since $a + b = e + f$, the relation is transitive. ‖ ∻Answer∻ ⚹ $R$ is an equivalence relation. ⚹ ∻Key Concept∻ ⚹Equivalence Relation⚹ ∻Explanation∻ ⚹An equivalence relation must be reflexive, symmetric, and transitive. We have shown that $R$ satisfies all three properties.⚹◊What are the three properties that need to be proven in order to show that $R$ is an equivalence relation⍭ Generate me a similar question◊

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