Relations on Sets

1
Section 10.1

• Relations on Sets

2
What Is a Relation?

• There are lots of kinds of relationships in the
  world some between people, some between
  objects.
• Were going to discuss the relationship between
  objects in sets.
• Well define a relationship.
• Then we can figure out whether two objects have
  the relationship were looking for or not.

3
Example of a Relation

• Let A 1, 3, 5 and B 2, 4, 6. Let us say
  that an element x in A is related to an element y
  in B if x lt y. Well write x R y as shorthand
  for x is related to y. Then
• 1 lt 2, so 1 R 2
• 1 lt 4, so 1 R 4
• 1 lt 6, so 1 R 6
• and so on..

4
Example of a Relation

• We use x R y to say that x is not related to y in
  the way we desire. So here,
• 3 R 2 because 3 lt 2
• 5 R 2 because 5 lt 2
• 5 R 4 because 5 lt 4
• and so on..

5
Example of a Relation

• Recall that the Cartesian product
• A x B (a, b) a ? A and b ? B.
• The set of ordered pairs (a, b) such that a R b
  is a subset of A x B
• (1, 2), (1, 4), (1, 6), (3, 4), (3, 6), (5, 6).

6
Definition of Relation

• Let A and B be sets. A relation R from A to B is
  a subset of A x B.
• Given an ordered pair (x, y) in A x B, x is
  related to y by R, written x R y, if, and only
  if, (x, y) is in R.
• Symbolically, x R y ? (x, y) ? R and x R y ? (x,
  y) ? R
• In our earlier example, the relation R would be
  the set
• (1, 2), (1, 4), (1, 6), (3, 4), (3, 6), (5, 6).

7
Another Relation

• Let A and B be Z and define a binary relation O
  from A to B as follows
• for all (m, n) ? Z x Z, m O n ? m-n is odd.
• In other words,
• O (m, n) m and n are integers and m-n is odd
• Then 3 O 0, 2 O 5, and 4 O 3, but 4 O 2.

8
Another Relation

• Let ? x, y and define a binary relation R
  from ?3 to ?3 as follows For all strings s and t
  in ?3, s R t ? the last character of s is the
  same as the last character of t.
• xyy R yxy
• xyx R xxx
• xyx R xyy

9
Functions and Relations

• A function F from a set A to a set B is a
  relation from A to B that satisfies the following
  two properties
• For every element x in A, there is an element y
  in B such that (x, y) ? F.
• For all elements x in A and y and z in B, if (x,
  y) ? F and (x, z) ? F, then x z.

10
So.What Is This New Definition?

• Its really the same definition that we had for
  functions before recast in terms of relations.
  It is mainly done just to point out that a
  function is only one type of relation.
• Everything that was a function before you saw
  this definition is still a function.
• If something was not a function before, then it
  is not a function now.

11
The Inverse of a Relation

• Here we dont need the same sort of restrictions
  we see with functions (where you need a bijection
  in order to have an inverse).
• Why not? If were looking at a relation from A
  to B, then
• not every element x of A needs to have an element
  y such that x R y and
• any element x in A can be related to more than
  one element of B.

12
The Inverse of a Relation

• Let R be a relation from A to B. Define the
  inverse relation R-1 from B to A as follows
• R-1 (y, x)? B x A(x, y) ? R.

13
Example

• Find the inverse of the relation we considered at
  the beginning of class. So. let A 1, 3, 5
  and B 2, 4, 6, and let us say that x R y if x
  lt y. Recall that
• R (1, 2), (1, 4), (1, 6), (3, 4), (3, 6), (5,
  6).
• Then R-1 is a relation from B to A defined as
  follows y R x if y gt x. Thus,
• R-1 (2, 1), (4, 1), (6, 1), (4, 3), (6, 3),
  (6, 5).

14
N-ary Relations and Relational Databases

• Given sets A1, A2, , An, an n -ary relation R on
  A1 x A2 x x An is a subset of A1 x A2 x x An.
• These are used to keep track of information that
  has more than one component in databases (i.e.,
  records at a hospital).

15
What Is Coming Up

• On Fridays class, we will cover Section 10.2
  reflexivity, symmetry, and transitivity.
• On Mondays class, we will have a review session
  for the final (our final is the 29th from
  8-1050). I will include suggested problems for
  Chapter 10 in the review material. A review
  sheet will be posted to our web site.
• On Wednesdays class and next Fridays class if
  necessary, we will cover Section 10.3
  (equivalence relations).
• With any remaining time, I will discuss how this
  class ties into CS3500 (Theory I).