Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Equivalence relations “are" partitions. Let S be a set, and RC S × S be an equivalence relation on S, i.e. RC S × S is a relation between S and S, and Ris • reflexive, i.e. for all s € S, (s, s) ≤ R, • symmetric, i.e. and for all (8₁, 82) € S × S, (S1, S2) ≤ R transitive, i.e. ($2,81) € R, for all (8₁, 82) € S × S for all (s2, 83) € S × S, (81, 82) € R and (82, 83) € R ($1,$3) € R. (17) (18) (19) Then, prove the collection of equivalence classes of S via the (equiva- lence) relation R partitions S1
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Transcribed Image Text:Equivalence relations “are" partitions. Let S be a set, and RC S × S be an equivalence relation on S, i.e. RC S × S is a relation between S and S, and Ris • reflexive, i.e. for all s € S, (s, s) ≤ R, • symmetric, i.e. and for all (8₁, 82) € S × S, (S1, S2) ≤ R transitive, i.e. ($2,81) € R, for all (8₁, 82) € S × S for all (s2, 83) € S × S, (81, 82) € R and (82, 83) € R ($1,$3) € R. (17) (18) (19) Then, prove the collection of equivalence classes of S via the (equiva- lence) relation R partitions S1
14Recall an equivalence class of S via the (equivalence) relation R is a subset of S, consisting of elements, all of whom are related to one another (by R). More precisely (at the expense of intuition!), an equivalence class of S via the (equivalence) relation R is a subset (20) TCS such that there exists t € S such that T = :{s € S|(s, t) ≤ R}. We may choose, for each equivalence class T of S via the (equivalence) relation R, a repre- sentative t € T, i.e. a specific element t € T to which all other elements of T are equivalent via R, i.e. a specific element t € T such that for all s € T, (s, t) € R. Having selected a representative, t, for an equivalence class T of S via the (equivalence) relation R, we may denote T instead, more suggestively, by [t], which signifies “all of the elements of S which are equivalent to t via the (equivalence) relation R.”" - For example, let S := Z × Z − {(0,0)} =: Zײ – {(0,0)} and R := {((x, y), (z, w)) € Sx S|xw = yz} C S x S. We have shown (and you should check once more!) RC S x S is an equivalence relation on S. The set T := {..., (-3, −6), (−2, −4), (−1, −2), (1, 2), (2, 4), (3, 6), ...} (21) is an equivalence class of S via the (equivalence) relation R, and happens to be the equiv- alence class of t := (1, 2) (ort := (−3, −6), or t := (−2, −4), or t := (−1, −2), or t := : (2,4), or t := (3,6), etc.) Remind yourself that the equivalence classes of S via the (equivalence) relation in this specific example do indeed partition S (into a collection of disjoint lines).
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Transcribed Image Text:14Recall an equivalence class of S via the (equivalence) relation R is a subset of S, consisting of elements, all of whom are related to one another (by R). More precisely (at the expense of intuition!), an equivalence class of S via the (equivalence) relation R is a subset (20) TCS such that there exists t € S such that T = :{s € S|(s, t) ≤ R}. We may choose, for each equivalence class T of S via the (equivalence) relation R, a repre- sentative t € T, i.e. a specific element t € T to which all other elements of T are equivalent via R, i.e. a specific element t € T such that for all s € T, (s, t) € R. Having selected a representative, t, for an equivalence class T of S via the (equivalence) relation R, we may denote T instead, more suggestively, by [t], which signifies “all of the elements of S which are equivalent to t via the (equivalence) relation R.”" - For example, let S := Z × Z − {(0,0)} =: Zײ – {(0,0)} and R := {((x, y), (z, w)) € Sx S|xw = yz} C S x S. We have shown (and you should check once more!) RC S x S is an equivalence relation on S. The set T := {..., (-3, −6), (−2, −4), (−1, −2), (1, 2), (2, 4), (3, 6), ...} (21) is an equivalence class of S via the (equivalence) relation R, and happens to be the equiv- alence class of t := (1, 2) (ort := (−3, −6), or t := (−2, −4), or t := (−1, −2), or t := : (2,4), or t := (3,6), etc.) Remind yourself that the equivalence classes of S via the (equivalence) relation in this specific example do indeed partition S (into a collection of disjoint lines).
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