Relations II

1. Relations II • Examples: on the set of positive integers is bigger than is bigger than or equal to is a factor of mRn iff mn is square mRn iff m+n is a multiple of 3 mRn iff m, n have same number of factors

2. 1 2 3 4 …. Example 1 mRn iff m > n Examples: 3 R 1, 56 R 45 (substitute in for m and n) Directed Graph:

3. reflexive Is “is bigger than” a reflexive relation? (do elements of the set relate to themselves?) no proof? Give an example of an element which doesn’t relate to itself

4. A non-reflexive proof Proposition: The relation R is not reflexive Proof: It is not the case that 13 is bigger than 13, so 13 R 13 Therefore, it is not true that for all x, x R x.

5. symmetric Is the relation “is bigger than” symmetric? (do all arrows have reverse arrows?) no. proof: find one arrow whose reverse isn’t included

6. Proof for non-symmetry Proposition: R is not symmetric Proof: It is the case that 23 is bigger than 12 but it is not that case that 12 is bigger than 23. So 23 R 12, but 12 R 23. It is not the case that for all x, y, xRy implies that yRx

7. transitivity Is “is bigger than” transitive? (can you shortcut triangles?) On my directed graph, all arrows pointed left and anything which goes left was there, so all triangles could be shortcut Yes, the relation is transitive

8. Proof of transitivity Proposition: R is transitive. Proof: Take any positive integers x, y, z which have xRy and yRz. ie. x is bigger than y and y is bigger than z. So x must be bigger than z, and xRz. So, for all choices of x,y,z with xRy and yRz, we can show than xRz.

9. antisymmetry Is “is bigger than” antisymmetric? (you never see arrows with their reverse?) This seems to be the case - intuitively using the “points left” idea yes, the relation is antisymmetric

10. Proof of antisymmetry Proposition: R is antisymmetric Proof: Take any x, y with xRy and yRx. xRy means that x is bigger than y, and yRx means that y is bigger than x. This can’t ever happen, and it’s not possible to find such an x,y. It is the case that xRy , yRx implies x = y (because F implies anything)

11. total Is “is bigger than” total? (can you compare any two elements, one way round or the other?) Given two numbers, can we always say that one is bigger than the other? (no) the relation is not total

12. Proof of non-totality Proposition: R is not total Proof: 13 is not bigger than 13, so 13 R 13 So if x =13, and y =13, x R y and y R x This means that it’s not true that for all choices of x,y, either xRy or yRx.

13. partial order Is “is bigger than” a partial order? (is the relation reflexive, anti-symmetric and transitive?) no - it’s not reflexive (and I don’t need to check the other two conditions now) The relation is not a partial order

14. Proof of not - partial order Proposition: R is not a partial order Proof: A partial order is a reflexive, anti-symmetric and transitive relation. But we have shown earlier that R is not reflexive.

15. mRn iff mn is square start with a directed graph: 1 2 3 4 5 6 7 8 9

16. mRn iff mn is square • reflexive? yes • symmetric? yes • transitive? not sure - do some examples… we need to choose x, y, z (could be equal) with xRy and yRz eg: 1R1 and 1R4, and 1R4 eg: 4R1 and 1R9, and 4R9 guess yes

17. Proof for reflexive property Proposition: R is a reflexive relation. Proof: Take x, any positive integer. The product of x with itself is a square, xx is square, so xRx.

18. Proof for symmetry Proposition: R is a symmetric relation. Proof: Take x,y any positive integers with xRy. This means that the product xy is a square. But yx = xy, so yx must also be a square number, and yRx. For all x, y, with xRy, we also have yRx.

19. Proof for transitivity Proposition: R is a transitive relation. Proof: Take x,y, z any positive integers with xRy and yRz. This means that xy is a square and yz is a square. Let xy = p2 and yz = q2. so xz is a square and xRz.

20. Proof for equivalence relation Proposition: R is an equivalence relation. Proof: An equivalence relation is a relation which is reflexive, symmetric and transitive. We have shown that R has these three properties, so R is an equivalence relation.

21. Relations on sets • Examples: on the set of sets is a subset of has non-empty intersection with has a bijection to

22. Relations on sets A~B iff there is some bijective function from A to B Show that {1, 2, 3} ~ {a, b, c}. Show that {1,7,6,2} ~ {p, q}.

23. Equivalence relations and partitions The directed graph of an equivalence relation looks like this: The set splits into subsets which are completely interrelated - each element is in an interrelated subset This is a partition of the set

24. Equivalence classes Given an equivalence relation R on set A, and an element a of A, the equivalence class of a in A is the set of elements of A which are related to a.

25. The equivalence class of a for any a in A, a

26. The equivalence class of a [a] is a subset of A [a]

27. The equivalence class of b for any b in A, b

28. Equivalence relations and partitions [b] is a subset of A [b]

29. Equivalence relations and partitions in this example, [c] = [b] b c

30. Equivalence relations and partitions in this example, [a] = [d] d a

31. Equivalence classes Proposition: If R is an equivalence relation then [a] = [b] iff aRb. Proof: (needs two sections) First show that if [a] = [b] then aRb. This is because b is an element of [b], because R is reflexive. And [b] = [a], so b is in [a]. But , so we must have aRb. continued….

32. Equivalence classes Proposition: If R is an equivalence relation then [a] = [b] iff aRb. Proof: (continued) Now show that if aRb then [a] = [b]. Assume aRb. To show the subsets [a] and [b] are equal, show one contains the other and v.v. To show: [a] is a subset of [b] To show: elements of [a] must be in [b] ctd…..

33. Equivalence classes Proposition: If R is an equivalence relation then [a] = [b] iff aRb. Proof: (continued) aRb implies that elements of [a] must be in [b]: Assume aRb and that c is in [a], ie. aRc. bRa and aRc, so bRc, and c is in [b]. Now show that all elements of [b] are in [a] Assume aRb and that d is in [b], ie. bRd. R is transitive, so aRd, and d is in [a].

34. mRn iff mn is square • is an equivalence relation What is [1]? What is [2]? What is the partition of the set of positive integers?

35. Size of sets A~B iff there is some bijective function from A to B is an equivalence relation. List some elements of [{1,2}]. List some elements of [the set of integers].