Relations on Sets

Relations on Sets - Math 100 Relations 1. Relations on Sets...

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Math 100 Relations Martin H. Weissman 1. Relations on Sets 1.1. Examples of Relations. Let S be a set. A relation on S is a subset of S 2 . In other words, a relation is a set of ordered pairs, whose entries are elements of S . If R is a relation on S , and ( a,b ) S 2 , then we say that “ a is related to b ” if ( a,b ) R . Let’s look at a simple example: Let S = { a,b } . Let R = { ( a,a ) , ( b,b ) } . Then R is a relation on S . In this case, the following two sentences are true: a is related to a . b is related to b . There are no other relations among the elements of S (for example, a is not related to b ). The relation R is called equality , because one element of S is related to another element of S if and only if they are equal to each other. Here’s another example: Let S = { a,b } . Let R = { ( a,b ) , ( b,a ) } . Then R is a relation on S . In this case, the following two sentences are true:
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