DISCRETE MATHEMATICS

1. DISCRETE MATHEMATICS • K.H. Rosen, Discrete Maths and its applications, McGraw-Hill. • Course covers introduction to set theory, functions, relations, logic, graphical representation.

11. 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.

12. 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.

13. 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.

14. 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}.

15. Injective,surjective,bijective • A function is injective if f(x)=f(y) implies 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)

16. 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)

17. Exercise • Consider two finite sets A and B. Evaluate: • a) The number of functions from A to B. • b) The number of injections from A to B. • c) The number of bijections from A to B.

18. Exercise- find examples of functions from N to N satisfying: • a) An injection but not surjection • b) A surjection but not injection • c) A bijection • d) neither a surjection or injection

19. 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 there is no more than one 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

20. 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 there is an element a in A for every b in B with f(a)=b, so f is surjective. Since f injective and surjective it is also bijective

26. 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 …’

27. 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.

28. 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 Imperial College.’ OR ‘this lecture course is not given at Imperial College.

29. 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’.

30. 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’

31. 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..

32. 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.

33. 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

34. 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

35. 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’

36. 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 →∨∧¬

37. 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‘

38. PREDICATE LOGIC • Propositional logic is limited, eg how do we express ‘All Maths undergrads are clever’ in terms of the cleverness of particular maths undergrads? 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

39. 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

40. 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)

41. 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))

42. 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)