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ma103blect3 - Applied Algebra Lecture 3 Relations Functions...

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1 Applied Algebra Lecture 3 - Relations, Functions and the Pigeonhole Principle Relations Defn. A (binary) relation R from set A to set B is a subset of the set A × B={(a,b}|a A,b B}. We write a R b if (a,b) R. A relation R on A is a subset of A × A. Examples. 1) A relation on is < that is the set of pairs (a,b) in × with a<b. 2) Another relation on is divisibility; i.e., the pairs (a,b) in × with a divides b. 3) Congruence (mod m) gives another relation on A=B= ; i.e., a b(mod m). Defn. A relation ~ on a set S is an equivalence relation iff it has the following 3 properties. Let’s write a~b instead of (a,b) the relation ~. 1) a~a for all a S ( reflexivity ) 2) a~b b~a ( symmetry ) 3) a~b and b~c a~c ( transitivity ) Let’s figure out whether the 3 relations in the preceding examples are equivalence relations on . 1) a<b is not an equivalence relation since it is not reflexive or symmetric. 2) a|b is not an equivalence relation since it is not symmetric.
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