Lets define a relation $R$ on $\mathcal{P}(Z)$ by $(S, T) ∈ R$ if and only if $S ⊆ T$. Which of the following properties does $R$ exhibit? (a) reflexivity, (b) transitivity, (c) symmetry, (d) anti-symmetry.
<aside> 💡 (a) Reflexivity: R is reflexive since for any set S ⊆ P(Z), S ⊆ S. (b) Transitivity: R is transitive since for any sets S, T, U ⊆ P(Z), if S ⊆ T and T ⊆ U, then S ⊆ U. (c) Symmetry: R is not symmetric since for any sets S, T ⊆ P(Z), if S ⊆ T, then T ⊈ S. (d) Anti-symmetry: R is anti-symmetric since for any sets S, T ⊆ P(Z), if S ⊆ T and T ⊆ S, then S = T.
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Consider a relation $R$ on the set of ordered pairs of integers defined by $\Big((a, b),(c, d)\Big)∈ R$ if and only iff $ad = bc$. Which properties does $R$ exhibit?
(a) reflexivity, (b) transitivity, (c) symmetry, (d) anti-symmetry.
Hint: This is the same test you would perform to check whether $\frac{a}{b} =\frac{c}{d}$ if $\frac{a}{b}$ and $\frac{c}{d}$ are rational numbers.
<aside> 💡 (a) Reflexivity: R is reflexive since for any integer a, R does not contain the pair ((a, a), (a, a)). (b) Transitivity: $bd\ne0$ (c) Symmetry: R is symmetric since for any pairs of ordered integers (a, b), (c, d), if ((a, b), (c, d)) ∈ R, then ((c, d), (a, b)) ∈ R only if a = c and b = d. (d) Anti-symmetry: R is not anti-symmetric since for any pairs of ordered integers (a, b), (c, d), if ((a, b), (c, d)) ∈ R and ((c, d), (a, b)) ∈ R, then (a, b) ≠ (c, d).
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