Let R_1 and R_2 be two relations in a set A. Let R be the new relation in A defi
ID: 3108978 • Letter: L
Question
Let R_1 and R_2 be two relations in a set A. Let R be the new relation in A defined as follows: x R y if x R_1 y and x R_2 y. (Typically we denote R by R_1 intersection R_2 or by R_1 logicaland R_2.) a) Prove that if R_1 is reflexive or R_2 is reflexive, then R is not necessarily reflexive. b) Prove that if R_1 is symmetric or R_2 is symmetric, then R is not necessarily symmetric. c) Prove that if R_1 is antisymmetric or R_2 is antisymmetric, then R is antisymmetric. d) Prove that if R_1 is transitive or R_2 is transitive, then R is not necessarily transitive.Explanation / Answer
clearly R is a relation which contans all the ordered paires which are common in both R1 and R2
(a) so if either R1 or R2 reflexive it is not guaranty that all the ordered pairs (a,a) for all a belongs to A will not be in R so R not necessarily reflexive.
(b) if either R1 or R2 are symmetric it contains all the orderepairs of the form when ever (a,b) belongs to either R1 or R2 , (b,a) may not be in R so R not necessarily symmetric
(c) if either R1 or R2 are anti symmetric clearly R is antisymmetric, since R has ordered pairs which has common in R1 and R2
(d) similiar case in the case of transitive
in (a), (b), (d), R is symmetric, transitive, reflexive when R1 and R2 are symmetric, transitive, reflexive, but given as R1 or R2
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