1. Binary Relations Binary Relations on Real Numbers

2. 10/7/2013 Binary Relations 2 2 Operations and Relations Whats the difference ? Binary relation ~ relates one number with another: x ~ y defines an ordered pair in a set of ordered pairs relating x to y in some way Binary operator ¤ acts on two elements to produce a third … … so that x ¤ y = z Arithmetic Axioms

3. 10/7/2013 Binary Relations 3 3 Operations and Relations Examples Binary relation: 1. x ~ y if and only if x is a prime that divides y evenly 2. x < y if and only if x is less than y Binary operation: 1. x + y = z 2. x y = z Arithmetic Axioms a set of ordered pairs a set of ordered triples

4. 10/7/2013 Binary Relations 4 4 Equivalence Relations:, =, Identity Relation: Used to identify or define operations Examples: x – y x + (-y) x / y x y –1 Used to identify equivalence classes Example: 10 2 mod 8 18 mod 8 26 mod 8 2 + 8k for some integer k Equivalence and Order

5. 10/7/2013 Binary Relations 5 5 Equivalence Relations:, =, Non-Equality Relation: Declaration of non-interchangeability Examples: (3 + 4) 5 (3 + 4) (9 + 2) 1 0 Equivalence and Order

6. 10/7/2013 Binary Relations 6 6 Equivalence Relations:, =, Equality Relation: = Declaration of interchangeability … if x = y then x and y are interchangeable Examples: (3 + 4) = 7 (3 + 4) = (9 – 2) (x 2 – 1) = (x – 1)(x + 1) (x 2 – 1) = 24 Equivalence and Order

7. 10/7/2013 Binary Relations 7 7 Properties of = Relation Three primary properties: Reflexivity: x = x Symmetry: x = y if and only if y = x Transitivity: if x = y and y = z then x = z Equivalence and Order

8. 10/7/2013 Binary Relations 8 8 Order Relations Four relations:, Less Than: < x less than y in numerical value, i.e. to the left of y on the real number line, is written x < y Less Than or Equal: Equivalence and Order

9. 10/7/2013 Binary Relations 9 9 Order Relations Four relations:, Less Than or Equal: If x y then either x < y or x = y … but not both! Greater Than: > Equivalence and Order Why ? Question: reflexive ? symmetric ? transitive ? Is

10. 10/7/2013 Binary Relations 10 Order Relations Four relations:, Greater Than: > If x is greater than y in numerical value, i.e. to the right of y on the real number line, then x > y Greater Than or Equal: Equivalence and Order Question: reflexive ? symmetric ? transitive ? Is >

11. 10/7/2013 Binary Relations 11 Order Relations Four relations:, Greater Than or Equal: If x y then either x > y or x = y … but not both! Equivalence and Order Question: reflexive ? symmetric ? transitive ? Is Why ?

12. 10/7/2013 Binary Relations 12 Trichotomy of the Real Numbers If x and y are any real numbers, then exactly one of the following holds: x = y x < y x > y The Trichotomy Law Question: Why are and not included ?

13. 10/7/2013 Binary Relations 13 Think about it !