Transcription of Equivalence Relations - Mathematical and Statistical Sciences
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Equivalence Relations - Mathematical and Statistical Sciences
Equivalence Relations Definition An Equivalence relation on a set S, is a relation on S which is reflexive, symmetric and transitive. Examples: Let S = and define R = {(x,y) | x and y have the same parity}. , x and y are either both even or both odd. The parity relation is an Equivalence relation. 1. For any x , x has the same parity as itself, so (x,x) R. 2. If (x,y) R, x and y have the same parity, so (y,x) R. 3. If (x,y) R, and (y,z) R, then x and z have the same parity as y, so they have the same parity as each other (if y is odd, both x and z are odd; if y is even, both x and z are even), thus (x,z).
1) a is an upper bound for B, and 2) a R x for every upper bound x for B. Example: Reals with the usual ordering. B = {x | 5 < x < 7 } 8, 7.5, 11, 7.00001 are all upper bounds of B. 7 is also an upper bound, and 7 ≤ any of the upper bounds, so 7 …
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