1. EE1J2 – Discrete Maths Lecture 7
What is a set?
What is a relation on a set?
What is a function between sets?
See Truss, Chapter 3

2. Some Example Sets {a,b,c,d}, {0,1,2,3}
{0,1,2,3,…} - the set ℕ of Natural Numbers
{0,2,4,6,…} the set of even natural numbers
{2n : n=0,1,2,3,…}
{n : n=2m, m ℕ}
{…,-3,-2,-1,0,1,2,3,…} - the set ℤ of Integers
{ : n, m  ℤ} – the set ℚ of rational numbers
The set ℝ of real numbers
 The empty set

3. Some more sets
The set ℂ of complex numbers x+jy, where x,y are members of ℝ and j2=-1
The closed interval [a,b] = {x: ax b}
The open interval (a,b) = {x: a<x <b}
The half-open interval [a,b)={x:ax <b} (or (a,b]={x:a<x  b}
The infinite intervals (-,), [a, ) and (-,b] are defined similarly

4. How do we define a Set? Axiom of Extensibility Axiom of Comprehension
a set is defined completely by its members
E.g: {0,1,2,3}
Axiom of Comprehension
any property defines a set – the set of all elements which satisfy that property
E.g: {n : n=2m, m ℕ}

5. Notation Let S be a set. If x is a member of S write x  S
1  {0, 1, 2, 3}
4  {0, 1, 2, 3}
Let T be another set. If each member of T is a member of S then T is a subset of S and we write T  S. Sometimes say T is contained in S
If there is an element x such that x  S but x  T then T is a proper subset of S and we write T  S

6. Notation continued {0,1}  {0, 1, 2, 3} and {0,1}  {0, 1, 2, 3}
{0, 1, 2, 3}  {0, 1, 2, 3}
but {0, 1, 2, 3}  {0, 1, 2, 3}
0  {0, 1, 2, 3}
0  {0, 1, 2, 3}
{0}  {0, 1, 2, 3}
  {0, 1, 2, 3}, but   {0, 1, 2, 3}
In fact, for any set S,   S

7. Ambiguity {a,b}={b,a}={a,a,b,b,b}={a,b,a,b,b,a}
In other words, order and repetition are not important.
{3,5}={n  ℕ: (n is prime)(n<6)}
= {x  ℤ: x2-8x+15=0}

8. Union, Intersection, Difference
Let S and T be sets
The union of S and T is the set
S  T = {x: (xS)(x T)}
The intersection of S and T is the set
S  T = {x: (xS)(x T)}
The difference of S and T is the set
S – T = {x: (xS)(x T)}
S – T is sometimes called the (relative) complement of T (in S) and written S \ T

9. Cardinality Let S be a set
The cardinality of S is the number of elements in S, written |S|
For finite sets the meaning of |S| is clear (e.g. |{0,1,2,3}|=4).
Cardinality also be extended to infinite sets.

10. The Power Set
Let S be a set. The Power Set of S is the name given to the set of all subsets of S. It is normally denoted by P(S)
Formally:
P(S) = {T: T  S}
Note that |P(S)|=2|S|

11. Proofs in Set Theory
To prove that two sets S and T are equal it is necessary to show that they have the same elements
One strategy is to show S  T and T  S
Each member of S is a member of T, and
Each member of T is a member of S
An alternative is to use the equivalences from propositional logic

12. Proof based on PL Let A and B be sets. Prove that
(AB)-(AB) = (A-B) (B-A)
Proof
(AB)-(AB)
={x:(xA  xB)(xA  xB)}
={x:(xA  xB) (xA  xB)}
={x:(xA  xA)}{x:(xA  xB)}
{x:(xB  xA)}  {x:(xB  xB)}
= {x:(xA  xB)} {x:(xB  xB)}
= (A-B) (B-A)

13. Proof using Venn DiagramsB X
X-A
A  B
A  B
A-B
B-A
(A-B)  (B-A)
(A  B) - (A  B)

14. Ordered Sets and n-tuples
{a,b} = {b,a} – order is not important
If order is important, we use (,) not {,}
So, (a,b)  (b,a)
(a,b) called an ordered pair
More generally, (a1,a2,…,an) is called an (ordered) n-tuple
(a1,a2,…,an,…), or (an)nℕ is a (infinite, ordered) sequence

15. Cartesian Product Let A and B be sets.
The Cartesian product of A and B is the set AB = {(a,b): (aA)(bB)}
| AB| = |A||B| (hence ‘product’)
To see this, suppose A={a1,…,aN} and B={b1,…,bM}, then…

16. Cartesian Product

17. Relations
Suppose A = {0,1,2,3}. An example of a relation on A is ‘<’ (‘less than’)
This relation is defined by the set
R = {0<1, 0<2, 0<3, 1<2, 1<3, 2<3}
or, equivalently
R = {(0,1), (0,2), (0,3), (1,2), (1,3), (2,3)} AA
More generally, a relation on A is a subset R of A  A

18. Relations (continued)
Mathematicians sometimes use the symbol ‘~’ to denote a relation. I.e. if a,b A, and if (a,b) R we write a ~ b or aRb
The set of possible relations on A is equal to the set of all possible subsets of A  A
i.e. P (A  A)
The number of possible relations on A is therefore 2|A  A|
For example, suppose A = {a,b}

19. Relations (example) A = {a,b} A  A = {(a,a), (a,b), (b,a), (b,b)}
P(A  A ) = {, {(a,a)}, {(a,b)}, {(b,a)}, {(b,b)}, {(a,a),(a,b)}, {(a,a),(b,a)}, {(a,a),(b,b)}, {(a,b),(b,a)}, {(a,b),(b,b)}, {(b,a),(b,b)}, {(a,a),(a,b),(b,a)}, {(a,a),(a,b),(b,b)}, {(a,a),(b,a),(b,b)}, {(a,b),(b,a),(b,b)}, {(a,a), (a,b), (b,a), (b,b)}}
These are also the possible relations on A.

20. Relations (continued)
As with sets, there may be more than one way to define a given relation
E.g: Let A = {2, 6, 12}. For this set,
‘divides’ is the relation {(2,2),(2,6),(2,12),(6,6),(6,12),(12,12)}
‘is less than or equal to’ gives the same relation
We can describe either relation using a directed graph (or digraph)

21. Representation of a Relation as a Directed Graph2 6 12
A = {2, 6, 12}. a ~ b if and only if a  b

22. Functions
You probably have pre-conceived ideas of what a function is – f(x)=x2+2x+2, f(x)=sin(x), f(x)=exp(x),…
These are all functions which associate a member x of ℝ unambiguously with another member f(x) of ℝ. Remember, ℝ is the set of real numbers
They can all be written in set-theoretic notation as f = {(x,f(x)): x  ℝ}

23. Summary of Lecture 7 Introduction to sets Relations on sets
Axioms of extensibility and comprehension
Relationship with propositional logic
Venn diagrams
Relations on sets