Relations and Orders in Mathematics

1. Equivalence Relations Partial Orderings

2. A relation R on a set A is an equivalence relation if it has these properties: R is reflexive R is symmetric R is transitive. The simplest example is equality: a = a; if a = b, then b = a; if a = b and b = c, then a = c. On the set A = {a, ,b, c, d}, = is the relation {<a,a>, <b,b>, <c,c>, <d,d>}

3. On the set of people the relation Sib = “has the same parents as” is an equivalence: Anyone has the same parents as themselves (so sibling is not quite the same) If x has the same parents as y, then y has the same parents as x If x has the same parents as y and y the same as z, then x has the same as z. (if we allowed half-siblings this would not work) Common arithmetic example: On Z>=0, =6is the relation {x =6y if x = y mod 6} 0 =66 =612 =618, etc., 1 =67 =613 …, 2 =68 =614 …, 3 =69 =615 …, 4 =610 =616 …, 5 =611 =617 ….

4. Equivalence classes: The equivalence class for an equivalence relation, R, on a set A for an element a ∈ A, is {b ∈ A | aRb}. We write [a] for this equivalence class. For the relation Sib from earlier, [chris (me)] = {chris, jack, lizanne} For any person p, [p] = {p, any r where r is a sibling of p} For the relation =6, the set of all equivalence classes is { {0,6,12,…}, {1,7,11,…}, {2,8,14,…}, {3,9,15,…}, {4,10,16,…}, {5,11,17,…} } There are a finite number of equivalence classes (6), each of which is infinite. We can see that the set of all equivalence classes for a relation R on the set A is a partition of A (all equivalence classes are disjoint and A is their union)

5. On the set A = {a, b, c, d} we saw that = is the relation {<a,a>,<b,b>,<c,c>,<d,d>} So the set of equivalence classes is { {a}, {b}, {c}, {d} } Any partition of a set defines an equivalence relation on the set. For A above, consider { {a,b}, {c,d} } The relation that has these equivalence classes is R1 = {<a,a>, <a,b>, <b,a>, <b,b>, <c,c>, <c,d>, <d,c>, <d,d>} What would be the relation, R2, for the partition { {a,b,c}, {d} } ? R2 = {<a,a>, <a,b>, <a,c>, <b,a>, <b,b>, <b,c>, <c,a>, <c,b>, <c,c>, <d,d>}

6. Equivalence in propositional logic Two propositions, P and Q, are equivalent if they have the same truth table: PQ For example: (r -> s)  (~r v s) This is an equivalence relation over propositions: Reflexive: P  P Symmetric: if P  Q, then Q  P Transitive: if P  Q and Q  R, then P  R

7. For the equivalence  over the propositions with a given set of primitives {p, q, r,…} an equivalence class is a a set of propositions that all have the same truth table. In labs 6 and 7 we generated sequences of propositions that were all equivalent until we reached one in a standard CNF form. All these propositions would be in the same equivalence class. For example, the following are all in one equivalence class (with many others) ((p -> q) & (q -> r)) -> (p -> r) ~(~(p & ~q) & ~(q & ~r) & ~(~(p & ~r))) (p & ~q) | (q & ~r) | ~p | r (p | q | ~p | r) & (p | ~r | ~p | r) & (~q | q | ~p | r) & (~q | ~r | ~p | r)

8. Definition 8.14 (Partial Order) Let A be a set. A relation ≼ on A is a partial order if it has these properties: reflexive, antisymmetric, transitive A relation ≺ on A is a strict partial order if it is irreflexive, antisymmetric, and transitive

9. Examples: On Z, the relation <= is a partial order. On Z, the relation < is a strict partial order On Z+, the relation | (divides) is a partial order Given any set, say A = {a,b,c}, consider the relation ⊆ on P(A), the power set of A. It is a partial order. Similarly, the relation ⊂ on P(A) is a strict partial order.

10. Looking at partial orders graphically: Hasse diagrams Consider the the relation ⊆ on P(A) for A = {a,b,c}. {a,b,c} {b,c} {a,b} {a,c} {b} {c} {a} { } Rather then show all arrows for the relation, we show those necessary to define the relation by adding loops for reflexive and adding the transitive closure.

11. The Hasse diagram for the the relation ⊆ on P(A) for A = {a,b,c}. For a partial ordering, we assume that each element is related to itself. We assume there are no arrows between elements at the same level. We assume that if there is an arrow x to y and y to z, then <x,z> is in the relation {a,b,c} {b,c} {a,b} {a,c} {b} {c} {a} { }

12. Consider the relation | (divides) on a finite set A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} 12 8 9 10 15 4 6 14 2 13 3 5 7 11 1

13. Consider the relation divides on the set {z ∈ Z : z | 30} 30 15 6 10 3 5 2 1

14. We have seen in the Hasse diagrams that partial orders can have elements that are not comparable, such as {a,b} and {b,c} in the diagram for P(A). Neither is a subset of the other. When for a partial order <, either a < b or b < a, then they are comparable, otherwise they are incomparable. In the divides ordering, 5 and 12 are incomparable, but 5 and 30 are comparable. A (strict) partial order for which all elements are comparable is a (strict) total order.

15. The order <= on Z is a total order. The order < on Z is a strict total order. The Hasse diagram for a total order is boring, it is simply a row (or tower) of arrows.

16. Minimal and maximal elements. A minimal element in a partial order < on a set S is an element x ∈ S such that for any y ∈ S such that y != x then not( y < x ) A maximal element in a partial order < on a set S is an element x ∈ S such that thfor anyy ∈ S such that y != x then not( x < y ) Example: consider the divides relation on {2,3,4,5,6,7,8,9,10, 12} (no 11) 12 8 We can see from the Hasse Diagram that 2, 3, 5, 7 are minimal elements and 7, 8, 9, 10 and 12 are maximal 9 10 4 6 2 3 5 7

17. For a partial relation < on a set S, a minimum element is an element x ∈ S such that for all y ∈ S, x < y a maximum element is an element x ∈ S such that for all y ∈ S, y < x The previous example did not have a minimum or maximum. However the divisors of 30 with the | (divides) order has minimum 1 and maximum 30 30 15 6 10 3 5 2 1

18. The last theorem for the semester: Theorem 8.3 (Every partial order has a minimal/maximal element on a finite set) Let ≼ ⊆ A × A be a partial order on a finite set A. Then ≼ has at least one minimal element and at least one maximal element.