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# relationskey2 - McCombs Math 381 Relations on Sets Basic...

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McCombs Math 381 Relations on Sets 1 Basic Idea: Given a set A , a relation R defined on set A is a rule that describes how to compare the elements of set A to one another. In some cases, this rule will be described explicitly by some formula or equation. In other cases, this rule will be described by listing ordered pairs of set elements. Formal Definition: Given a set A , a relation R on set A is a set of ordered pairs comprised of the elements of set A . We use the notation: R = x , y ( ) : x ! A , y ! A { } . Important Notation: xRy corresponds to the ordered pair x , y ( ) Important Fact: Every relation R : A ! A is a subset of the set A ! A , i.e. R ! A " A . Relations Between Sets: Given sets A and B , a relation R from A to B is a set of ordered pairs comprised of the elements of the sets A and B . We use the notation: R = x , y ( ) : x ! A , y ! B { } Special Properties: Suppose R is a relation on set A . 1. R is reflexive means that for every element x ! A we have xRx . In other words, for every element x ! A , the ordered pair x , x ( ) ! R . 2. R is symmetric means that for every element x , y ! A we have xRy ! yRx . In other words, if x , y ( ) ! R , then y , x ( ) ! R . 3. R is transitive means that for every element x , y , z ! A , we have xRy ! yRz ( ) " xRz . In other words, if x , y ( ) ! R and y , z ( ) ! R , then x , z ( ) ! R . 4. R is irreflexive means that for every element x ! A we have x R x . In other words, for every element x ! A , the ordered pair x , x ( ) ! R .

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