Unformatted text preview: A . Prove or give a counterexample to each of the following assertions. (a) If R 1 and R 2 are symmetric, then R 1 ∪ R 2 is symmetric. [2] (b) If R 1 and R 2 are antisymmetric, then R 1 ∩ R 2 is antisymmetric. [2] (c) If R 1 ∪ R 2 is re³exive, then either R 1 or R 2 is re³exive. [2] (d) If R 1 ∩ R 2 is re³exive, then both R 1 and R 2 are re³exive. [2] 6. If A = { a, b, c, d, e, f, g } , determine the number of relations on A that are (a) re³exive and symmetric. [2] (b) re³xive and contain ( a, b ) and ( b, c ). [2] 7. Let R consists of all pairs ( x, y ) ∈ Z × Z such that x 2 − y 2 is divisible by 3. (a) Show that R is an equivalence relation. [3] (b) Determine the partition induced by R . [2] 1...
View
Full Document
 Spring '11
 HuangJing
 Math, Equivalence relation, Binary relation, Symmetric relation, Identity element

Click to edit the document details