Let N be the set of natural numbers and R be the relation on $N \times N$ defined by (a, b) R (c, d) iff ad = bc for all a, b, c, d $\in$ N. Show that R is an equivalence relation.

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$\\ $ Reflexive: $ \\ $ (a, b) $ \in \mathrm{N} \times \mathrm{N} \\ a \cdot b=b \cdot a \\ \Rightarrow (a, b) R(a, b) \\ $ R is reflexive $ \\\\ $ Symmetric: $ \\ $ (a, b),(c, d) $ \in N \times N \\ $ (a, b) R(c, d) $ \Rightarrow a \cdot d=b \cdot c \\ \mathrm{c} \cdot \mathrm{b}=\mathrm{d} \cdot \mathrm{a} \\ \Rightarrow(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b}) \\ \mathrm{R} \text { is symmetric } \\\\ $ Transitive:$ \\ (a, b),(c, d),(e, f), $ \in N \times N \\ $(a, b) R(c, d)$ \& $(c, d) R(e, f)$ \Rightarrow a \cdot d=b \cdot c \ \& \ c \cdot f=d \cdot e \\ a \cdot d \cdot c \cdot f=b \cdot c \cdot d \cdot e \Rightarrow a \cdot f=b \cdot e \\ (a, b) R(e, f)\\ R \text { is transitive } \\ $ \quad \mathrm{R} $ is reflexive , symmetric and transitive. Hence R is an equivalance relation$

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Let N be the set of natural numbers and R be the relation on defined by (a, b) R (c, d) iff ad = bc for all a, b, c, d N. Show that R is an equivalence relation.

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Let N be the set of natural numbers and R be the relation on $N \times N$ defined by (a, b) R (c, d) iff ad = bc for all a, b, c, d $\in$ N. Show that R is an equivalence relation.