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

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

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

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}

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

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

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}

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

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.

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|

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

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)

Proof using Venn Diagrams B X X-A A  B A  B A-B B-A (A-B)  (B-A) (A  B) - (A  B)

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

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…

Cartesian Product

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

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}

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.

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)

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

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}

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