Choose one of the following relations. Determine if it isReflexive, Antisymmetri

Question

Choose one of the following relations. Determine if it isReflexive, Antisymmetric, Symmetric, and Transitive. Be sure togive support for your conclusion to each part. R1 (a,b) ~ (c,d) iff a + d = b + c R2 a,b) ~ (c,d) iff ad = bc R3 a,b) ~ (c,d) iff a = c or b = d R4 a,b) ~ (c,d) iff ab > cd Choose one of the following relations. Determine if it isReflexive, Antisymmetric, Symmetric, and Transitive. Be sure togive support for your conclusion to each part. R1 (a,b) ~ (c,d) iff a + d = b + c R2 a,b) ~ (c,d) iff ad = bc R3 a,b) ~ (c,d) iff a = c or b = d R4 a,b) ~ (c,d) iff ab > cd

Explanation / Answer

Reflexive: (a,b) ~ (a,b) because ab = ba. Symmetric: (a,b) ~ (b,a) because ab = ab Transitive: (a,b) ~ (c,d) and (c,d) ~ (e,f) implies (a,b) ~(e,f) because ad = bc and cf = de. Multiplying the second one by b, weget
bcf = deb and plugging in the first one, (ad)f = deb adf - deb = 0 d(af - eb) = 0 So either d or af - eb equal zero. d can't be zero soaf-eb = 0, or af = eb. Hence, (a,b) ~ (e,f).

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