DISCRETE MATHEMATICS

• K.H. Rosen, Discrete Maths and its applications, McGraw-Hill.

• Course covers introduction to set theory, functions, relations, logic, graphical representation.

Functions

• Df A function is a rule which assigns to each member a of a set A a unique member b of set B. We write f: A→B. Alternatively we write f(a)=b. So far have been concerned with the case when A=B=R (the set of real numbers)

• Note for each element a of A an element b exists in B but not vice-versa.

Functions as relations

• Recall relation is a subset of AxB

• A function is a special case of relation R where (a,b) in R implies there is no (a,b’) in R unless b , b’ coincide and where there is (a,b) in R for any a in A.

• Thus for a function F the following statements are equivalent

• F(a)=b, aFb, (a,b) F.∈

Domain and Codomain of f: A→B

• A is the domain, B is the codomain of f

• Example let f be the function that rounds up a number to the nearest integer. Hence round(.1)=0, round (3.6)=4. So round R→Z.

• Hence domain is the set of real numbers whilst the codomain is the set of integers.

Image, pre-image and Range

• If f(a)=b we say b is the image of a under f. Likewise we refer to a as the pre-image of b. We also talk about the images and pre-images of subsets of A and B. For example if S is a subset of A the image of S is f(S)={f(s) | s S}. The ∈range of a function is the set f(A).

• Ex suppose f: R→R,f(x)=x², then image of 2 is 4, pre-image of 9 is 3, Range of f is {x R,|x>0}.∈

Injective,surjective,bijective

• A function is injective if f(x)=f(y) imples x=y. ( one to one mapping) Examples f(x)=x+1, f(x)=x² both defined on R are injective, not injective. But if we define only on positive real numbers both are injective.

• A function is surjective if for every element b of B there exists a in A with f(a)=b. ( mapping is onto)

Example

• f:R→R f(x)=x+1 is surjective f(x)=x² is not surjective since no x exists with f(x)=-1. But if we define on C the set of complex numbers both are surjective.

• Definition If f is surjective and injective we say it is bijective ( one to one and onto)

Inverse of a function

• Df The function f ¹:B→A is the inverse of f : A ⁻→B and has the property that f(f ¹(x))=x,:x A⁻ ∈

• Theorem The function f has an inverse iff f ¹ is ⁻bijective. To prove this think of f ¹ as the relation ⁻

{(b,a))},with (a,b) f B×A⋃ ∈ ⊆• Since f injective only element of A for each

element of B and since f surjective there is no less than one element of A for each element of B so f ¹ is a function⁻

Inverse

• Now show if f has an inverse it must be bijective• If f(a)=f(b)it follows that f ¹(f(a))=f ¹(f(b) so by ⁻ ⁻

definition of inverse we have a=b so that f is injective

• For any b in B a=f ¹(b) and then f(a)=f(f ¹(b))=b ⁻ ⁻so for there is an element a in A for every b in B with f(a)=b, so f is surjective. Since f injective and surjectiveit is also bijective

INTRODUCTION TO LOGIC

• What do we mean by logic?

• Oxford Dict. The systematic use of symbolic techniques and mathematical techniques to determine the forms of valid deductive argument.

• Thus logic is the common language by which we can demonstrate the validity of our reasoning …’

PROPOSITIONS AND NEGATIONS

• Proposition is a declarative statement that is true or false, but not both.

• Ex All Maths undergraduates wear sandals. False• All Maths undergraduates have tutors. True• If a relation is transitive then Rⁿ is transitive. True • Hopefully the Circle Line will be running tonight. Not a

proposition.• Propositional logic is the branch of logic dealing with

reasoning about propositions.• We will use symbols to denote propositions. Ex let p be

the proposition all Maths students wear glasses. Let p be the proposition that all EE1 students know how to wire a plug. False.

NEGATION AND USE OF PROPOSITIONS

• Use several propositions to build compound propositions

• Df Let p be a proposition. Then the proposition ‘ It is not the case that p’ is another proposition called the negation of p and written as ¬p. Ex This lecture course is given at Imperial College. Its negation is ‘ It is not the case that this lecture course is given at Impeiral College.’ OR ‘this lecture course is not given at Impeiral College.

Conjuncton and Disjunction

• Given propositions p and q the proposition ‘p and q’ written p q is true when both p and q are true and false ∧otherwise, it is called the conjunction of p and q.

• Given propositions p and q then the proposition ‘p or q’ denoted by p q is the proposition that is false when both ∨p and q are false and true otherwise. p q is called the ∨disjunction of p and q.

• Ex Suppose p is ‘Maths undergraduates love tofu’ and q is ‘Maths undergraduates are weird’

• Then p q is ‘ Maths undergrads either love tofu or are ∨weird or both’.

• p q is ‘maths undergrads like tofu and are weird’.∧

IMPLICATIONS

• When we say p implies q, written p→q, we mean the proposition which is false when p is true and q is false, and true otherwise. ( think …)

• When p→q we might say ‘ if p, then q’, ‘ p is sufficient for q’, ‘q if p’, ‘q is necessary for p’, ‘p only if q’

• Example Suppose p is ‘I revised’ and q is ‘I passed the exam’ Then p→q may be expresed as ‘if I revised I passed’ ‘I passed if I revised’ or ‘I revised only if I passed’

Implications

• There is no need for a relationship between the premise and conclusion. Example ‘If all Maths undergrads like tofu, then 1+1=2. True regardless of tofu.

• ‘If all Maths undergrads like tofu then 1+2=4’ True because not all undergrads like tofu.

• ‘If we are in london 1+2=4’ False because we are in London but 1+2 is 3.

• ‘If all Maths undergrads like tofu then 1+2=4’ True since not all maths students like tofu.

• Note that in English ‘implies’ can also mean causes so English considers the meaning of propositions whilst Logic considers whether they are true or false..

CONVERSE, CONTRAPOSITIV, AND INVERSE

• a iff b is the same as (a→b) (b→a) and is ∧written as a⇔b. Hence a⇔b is true when a and b are both true or when a and b are both false (and is otherwise false)

• We define b→a to be the converse of a→b.• The contrapositive of a proposition a→b is the

proposition ¬b→¬a.• The inverse of a→b is the proposition ¬a→¬b.

LOGICAL EQUIVALENCE

• A tautology is a compound proposition that is always true, eg p ¬p is a tautology.∨

• A contradiction is a compound proposition that is always false. Eg p ¬p is a contradiction.∧

• We say propositions p and q are logically equivalent, written as if p≡q, if p⇔q is a tautology. Eg ¬(p q)≡¬p ¬q.. De Morgans ∨ ∧Theorem

• We can express an implication in terms of a disjunction and a negation p→q≡¬p q∨

LOGIVAL EQUIVALENCE

• Hence an implication is logically equivalent to its contrapositive

• p→q≡¬p q≡q ¬p≡¬q→¬p∨ ∨• If I revised, I passed ≡ if I didn’t pass I

didn’t revise.

• Likewise its converse is logically equivalent to its inverse.

• q→p≡¬q p≡p ¬q≡¬p→¬q∨ ∨

LOGICAL EQUIVALENCE

• The most common mistake in logic is to assume an implication is logically equivalent to its inverse

• Examples: ‘if I revised I passed’ is not equivalent to ‘ if I didn’t revise I didn’t pass’

• ‘if I eat too much I will get fat’ is not equivalent to ‘if I don’t eat too much I will not get fat’

OPERATOR PRECEDENCE

• BASIC RULES

• a ¬b≡a (¬b) ∧ ∧• a b c≡a (b c) ∨ ∧ ∨ ∧• a b→c≡(a b)→c∨ ∨• Hence order of increasing precedence is

→ ¬∨∧

LOGIC AND ENGLISH LANGUAGE

• Language often ambiguous so try and identify the basic propositions and build into logical statements.

• I sell donuts and coffee ≡ ‘I sell coffee’ ^I sell donuts’• I am not good at golf≡¬(I am good at golf)• If its raining its not sunny≡raining→¬sunny• You can have chicken or fish ≡’you can have chicken’

’you can have fish’∧• Unless causes problems!!! I will play golf if it doesn’t rain

can be ¬(′it will rain')→'I will play golf‘• or ¬(′it will rain')⇔′I will play golf‘

PREDICATE LOGIC

• Propositional logic is limited, eg how do we express ‘All Maths undergrads are clever’ in terms of the cleverness of particular maths undergads? Or express ‘there is a maths undergrad wearing sandals’ in terms of whether each individual maths undergrad is wearing sandals

• We therefore generalize the idea of a proposition to a predicate, predicates take one or more variable as arguments

PREDICATE LOGIC

• Examples: Let P(x) be the statement “x>12” then P(1) is false, P(23) is true.

• Let P(x) denote the statement ‘x is a Professor in the Mathematics Department’ then P(Hall) is true but P(Limebeer) is false

• Thus P(x) is not true or false until we specify an argument

UNIVERSAL QUANTIFIERS

• Suppose P(x) is the predicate ‘has a heart’• Then we can discuss P(Hall), P(Limebeer) but

how can we say all humans have a heart. Predicates deal with this using quantification.

• Definition The universal quantification of P(x) is the proposition ‘P(x) is true for all values of x in the universe of discourse’ Hence in the universe of discourse consisting of all humans we would say the universal quantification of P(x) is true

• Notation We write xP(x) which we read as for ∀all x, P(x) for the universal quantification of P(x)

UNIVERSAL QUANTIFICATION

• Of course we also need to specify the universe of discourse but here can use set theory

• Suppose P(x) is the predicate x² 0, then ≽we can say x R P(x), ie for all real x ∀ ∈P(x)

• but ¬( x C P(x)) ∀ ∈

EXISTENTIAL QUANTIFIER

• Suppose P(x) is’ x is wearing a necklace’ then P(Florence) is the proposition ‘Florence is wearing a necklace’ but how do we say ‘there is an EE1 Student wearing a necklace’?

• Definition The existential quantification of P(x) is the proposition ‘there exists an element x in the universe of discourse such that P(x) is true’

• Notation we write xP(x) which we read as there ∃exist x such that P(x). Hence if J=set of all EE! Students we can write x JP(x)∃ ∈