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
R2 (a,b) ~ (c,d) iff ad = bc we know that ab = ba . so, ( a,b) ~ (a,b) R2 is reflexive. suppose (a,b) ~ (c,d). i.e. ad = bc. ==> cb = da provided the abelian property holds inthe given set where this relation is defined. ==> ( c , d ) ~ ( a, b) . so, R2 is symmetric. suppose (a,b) ~ (c,d) and (c,d) ~ ( e,f ) . that is ad = bc and cf = de. ==> ade = bce by multiplying e on the right . ==> a ( cf) = b (ce) ==> ( af ) c = ( b e ) c ==> af = be ==> ( a, b ) ~ ( e , f ) so, R2 is transitive R2 is an equivalence relation . please post the other questions in the next post.