1 / 13

Equivalence Relations

Equivalence Relations. Equivalence Relations. A relation R on A A is an equivalence relation when R satisfies 3 conditions:  x  A, xRx ( reflexive ).  x, y  A, xRy  yRx ( symmetric ).  x, y, z  A, (xRy  yRz)  xRz ( transitive ).

ginger
Download Presentation

Equivalence Relations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Equivalence Relations

  2. Equivalence Relations • A relation R on AA is an equivalence relation when R satisfies 3 conditions: • x  A, xRx (reflexive). • x, y  A, xRy  yRx (symmetric). • x, y, z  A, (xRy  yRz)  xRz (transitive). • How are the properties of an equivalence relation reflected in its graph representation?

  3. Examples? • Let P be the set of all human beings; R  PP. • Is R an equivalence relation if: • aRb when a is the brother of b? • aRb when a is in the same family as b? • R  N N. • xRy when x has the same remainder as y when they are divided by 5?

  4. Examples? • Let R2 = RR be the set of points in the plane. • Is E  R2 R2 when • E1 = {((x1,y1),(x2,y2)) | (x1,y1) & (x2,y2) are on the same horizontal line}? (E1 “partitions” the plane into horizontal lines.) • E2 = {((x1,y1),(x2,y2)) | (x1,y1) & (x2,y2) are equidistant from the origin}? (E2 “partitions” the plane into concentric circles.)

  5. Partitions • Let S be a set. A partition, (S), of S is a set of nonempty subsets of S such that: • Si  (S) Si = S (the parts cover S) • SiSj  (S), SiSj =  (the parts are disjoint) • Example: (check 2 conditions of partition): • “The same remainder when divided by 5” partitions N into 5 parts. • E1 partitions the plane into horizontal lines. • E2 partitions the plane into concentric circles.

  6. Partitions as Equivalence Relations • Let E  SS be an equivalence relation, and a S. • The equivalence class determined by a is: [a] = {b S | aEb}: the set of all elements of S equivalent to a. • Let P be the set of equivalence classes under E.

  7. aEb  [a] = [b] I.e., any member of [a] can name the class. Assume[a][b]: Without loss of generality, c [b] and c[a] (draw a Venn diagram) • aEb, (given); • bEc, (by assumption) • aEc, (E is transitive) • c [a] (definition of equivalence class). • Therefore, [a][b] is false.

  8. Equivalence classes partition S • To prove this, we must show that: (i) the union of all equivalence classes equals S; (ii) if a is not equivalent to b, then [a] [b] = . (i): Since E is reflexive, a S, there is some equivalence class that contains a: [a]. Therefore, a  S [a] = S.

  9. Equivalence classes partition S ... (ii): To show: For [a][b], [a] [b] =  • Assume not:  c  [a] [b]. • c  [a]  aEc which implies [a] = [c]; • c  [b]  bEc which implies [b] = [c]; • Therefore, [a] = [b], a contradiction. • Therefore, [a] [b] = . The set of equivalence classes partitions S.

  10. A partition defines an equivalence relation • Let (S) be a partition of S: • Si  (S) Si = S (the parts “cover” S) • SiSj  (S), SiSj =  (the parts are disjoint) • Define E = {(a,b) | a, b Si  (S)}. • Illustrate on blackboard. • Claim: E is an equivalence relation: E is reflexive, symmetric, & transitive.

  11. E is an equivalence relation (i): x S, xEx(reflexive): • Since (S) is a partition, everyx is in some part. • Every element x of S is in the same part as itself: xEx. (ii):x, y S, x Ey  y Ex (symmetric). • If x is in the same part as y, then y is in the same part as x.

  12. E is an equivalence relation ... (iii): x, y, z S, (x Ey  y Ez)  x Ez (transitive): • If x is in the same part as y and y is in the same part as z, then x is in the same part as z. E is an equivalence relation

  13. Equivalence relations: summary • Partitioning a set S is the same thing as defining an equivalence relation over S. • If E is an equivalence relation of S, the associated partition is called the quotient set of S relative to E and is denoted S/E.

More Related