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Let’s get started with. Logic !. Logic. Crucial for mathematical reasoning Important for program design Used for designing electronic circuitry Logic is a system based on propositions . A proposition is a (declarative) statement that is either true or false (not both).

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• Logic!

CMSC 203 - Discrete Structures

• Crucial for mathematical reasoning

• Important for program design

• Used for designing electronic circuitry

• Logic is a system based on propositions.

• A proposition is a (declarative) statement that is either true or false (not both).

• We say that the truth value of a proposition is either true (T) or false (F).

• Corresponds to 1 and 0 in digital circuits

CMSC 203 - Discrete Structures

• “Elephants are bigger than mice.”

Is this a statement?

yes

Is this a proposition?

yes

What is the truth value

of the proposition?

true

CMSC 203 - Discrete Structures

• “520 < 111”

Is this a statement?

yes

Is this a proposition?

yes

What is the truth value

of the proposition?

false

CMSC 203 - Discrete Structures

• “y > 5”

Is this a statement?

yes

Is this a proposition?

no

Its truth value depends on the value of y, but this value is not specified.

We call this type of statement a propositional function or open sentence.

CMSC 203 - Discrete Structures

• “Today is January 27 and 99 < 5.”

Is this a statement?

yes

Is this a proposition?

yes

What is the truth value

of the proposition?

false

CMSC 203 - Discrete Structures

• “Please do not fall asleep.”

Is this a statement?

no

It’s a request.

Is this a proposition?

no

Only statements can be propositions.

CMSC 203 - Discrete Structures

• “If the moon is made of cheese,

• then I will be rich.”

Is this a statement?

yes

Is this a proposition?

yes

What is the truth value

of the proposition?

probably true

CMSC 203 - Discrete Structures

• “x < y if and only if y > x.”

Is this a statement?

yes

Is this a proposition?

yes

… because its truth value does not depend on specific values of x and y.

What is the truth value

of the proposition?

true

CMSC 203 - Discrete Structures

• As we have seen in the previous examples, one or more propositions can be combined to form a single compound proposition.

• We formalize this by denoting propositions with letters such as p, q, r, s, and introducing several logical operators or logical connectives.

CMSC 203 - Discrete Structures

• We will examine the following logical operators:

• Negation (NOT, )

• Conjunction (AND, )

• Disjunction (OR, )

• Exclusive-or (XOR, )

• Implication (if – then,  )

• Biconditional (if and only if,  )

• Truth tables can be used to show how these operators can combine propositions to compound propositions.

CMSC 203 - Discrete Structures

• Unary Operator, Symbol: 

CMSC 203 - Discrete Structures

• Binary Operator, Symbol: 

CMSC 203 - Discrete Structures

• Binary Operator, Symbol: 

CMSC 203 - Discrete Structures

• Binary Operator, Symbol: 

CMSC 203 - Discrete Structures

• Binary Operator, Symbol: 

CMSC 203 - Discrete Structures

• Binary Operator, Symbol: 

CMSC 203 - Discrete Structures

• Statements and operators can be combined in any way to form new statements.

CMSC 203 - Discrete Structures

• Statements and operators can be combined in any way to form new statements.

CMSC 203 - Discrete Structures

• To take discrete mathematics, you must have taken calculus or a course in computer science.

• When you buy a new car from Acme Motor Company, you get \$2000 back in cash or a 2% car loan.

• School is closed if more than 2 feet of snow falls or if the wind chill is below -100.

CMSC 203 - Discrete Structures

• The statements  (PQ) and ( P)  ( Q) are logically equivalent, since they have the same truth table, or put it in another way, (PQ)  ( P)  ( Q) is always true.

CMSC 203 - Discrete Structures

• A tautology is a statement that is always true.

• Examples:

• R (R)

•  (PQ)  (P)( Q)

• A contradiction is a statement that is always false.

• Examples:

• R(R)

•  ( (P  Q)  ( P)  ( Q))

• The negation of any tautology is a contradiction, and the negation of any contradiction is a tautology.

CMSC 203 - Discrete Structures

• Definition: two propositional statements S1 and S2 are said to be (logically) equivalent, denoted S1  S2 if

• They have the same truth table, or

• S1  S2 is a tautology

• Equivalence can be established by

• Constructing truth tables

• Using equivalence laws (Table 5 in Section 1.2)

CMSC 203 - Discrete Structures

• Equivalence laws

• Identity laws, P  T  P,

• Domination laws, P  F  F,

• Idempotent laws, P  P  P,

• Double negation law,  ( P)  P

• Commutative laws, P  Q  Q  P,

• Associative laws, P  (Q  R) (P  Q)  R,

• Distributive laws, P  (Q  R) (P  Q)  (P  R),

• De Morgan’s laws,  (PQ) ( P)  ( Q)

• Law with implication P  Q  P  Q

CMSC 203 - Discrete Structures

• Show that P  Q  P  Q: by truth table

• Show that (P  Q)  (P  R)  P  (Q  R): by equivalence laws (q20, p27):

• Law with implication on both sides

• Distribution law on LHS

CMSC 203 - Discrete Structures

• Proposition

• Truth value

• Truth table

• Operators and their truth tables

• Equivalence of propositional statements

• Definition

• Proving equivalence (by truth table or equivalence laws)

CMSC 203 - Discrete Structures

• Propositional function (open sentence):

• statement involving one or more variables,

• e.g.: x-3 > 5.

• Let us call this propositional function P(x), where P is the predicate and x is the variable.

What is the truth value of P(2) ?

false

What is the truth value of P(8) ?

false

What is the truth value of P(9) ?

true

CMSC 203 - Discrete Structures

• Let us consider the propositional function Q(x, y, z) defined as:

• x + y = z.

• Here, Q is the predicate and x, y, and z are the variables.

true

What is the truth value of Q(2, 3, 5) ?

What is the truth value of Q(0, 1, 2) ?

false

What is the truth value of Q(9, -9, 0) ?

true

A propositional function (predicate) becomes a proposition when all its variables are instantiated.

CMSC 203 - Discrete Structures

• Let P(x) be a predicate (propositional function).

• Universally quantified sentence:

• For all x in the universe of discourse P(x) is true.

• Using the universal quantifier :

• x P(x) “for all x P(x)” or “for every x P(x)”

• (Note: x P(x) is either true or false, so it is a proposition, not a propositional function.)

CMSC 203 - Discrete Structures

• Example: Let the universe of discourse be all people

S(x): x is a UMBC student.

G(x): x is a genius.

• What does x (S(x)  G(x)) mean ?

• “If x is a UMBC student, then x is a genius.” or

• “All UMBC students are geniuses.”

• If the universe of discourse is all UMBC students, then the same statement can be written as

x G(x)

CMSC 203 - Discrete Structures

• Existentially quantified sentence:

• There exists an x in the universe of discourse for which P(x) is true.

• Using the existential quantifier :

• x P(x) “There is an x such that P(x).”

• “There is at least one x such that P(x).”

• (Note: x P(x) is either true or false, so it is a proposition, but no propositional function.)

CMSC 203 - Discrete Structures

• Example:

• P(x): x is a UMBC professor.

• G(x): x is a genius.

• What does x (P(x)  G(x)) mean ?

• “There is an x such that x is a UMBC professor and x is a genius.”

• or

• “At least one UMBC professor is a genius.”

CMSC 203 - Discrete Structures

• Another example:

• Let the universe of discourse be the real numbers.

• What does xy (x + y = 320) mean ?

• “For every x there exists a y so that x + y = 320.”

Is it true?

yes

Is it true for the natural numbers?

no

CMSC 203 - Discrete Structures

• A counterexample to x P(x) is an object c so that P(c) is false.

• Statements such as x (P(x)  Q(x)) can be disproved by simply providing a counterexample.

Statement: “All birds can fly.”

Disproved by counterexample: Penguin.

CMSC 203 - Discrete Structures

•  (x P(x)) is logically equivalent to x ( P(x)).

•  (x P(x)) is logically equivalent to x ( P(x)).

• See Table 2 in Section 1.3.

• This is de Morgan’s law for quantifiers

CMSC 203 - Discrete Structures

• A predicate can have more than one variables.

• S(x, y, z): z is the sum of x and y

• F(x, y): x and y are friends

• We can quantify individual variables in different ways

• x, y, z (S(x, y, z)  (x <= z  y <= z))

• x y z (F(x, y)  F(x, z)  (y != z)  F(y, z)

• Exercise: translate the following English sentence into logical expression

“There is a rational number in between every pair of distinct rational numbers”

CMSC 203 - Discrete Structures

• Propositional functions (predicates)

• Universal and existential quantifiers, and the duality of the two

• When predicates become propositions

• Nested quantifiers

• Logical expressions formed by predicates, operators, and quantifiers

CMSC 203 - Discrete Structures

• Mathematical Reasoning

CMSC 203 - Discrete Structures

• We need mathematical reasoning to

• determine whether a mathematical argument is correct or incorrect and

• construct mathematical arguments.

• Mathematical reasoning is not only important for conducting proofs and program verification, but also for artificial intelligence systems (drawing logical inferences from knowledge and facts).

CMSC 203 - Discrete Structures

• An axiom is a basic assumption about mathematical structured that needs no proof.

• We can use a proof to demonstrate that a particular statement is true. A proof consists of a sequence of statements that form an argument.

• The steps that connect the statements in such a sequence are the rules of inference.

• Cases of incorrect reasoning are called fallacies.

CMSC 203 - Discrete Structures

• A theorem is a statement that can be shown to be true.

• A lemma is a simple theorem used as an intermediate result in the proof of another theorem.

• A corollary is a proposition that follows directly from a theorem that has been proved.

• A conjecture is a statement whose truth value is unknown. Once it is proven, it becomes a theorem.

CMSC 203 - Discrete Structures

• Rules of inference provide the justification of the steps used in a proof.

• One important rule is called modus ponens or the law of detachment. It is based on the tautology (p  (p  q))  q. We write it in the following way:

• p

• p  q

• ____

•  q

The two hypotheses p and p  q are written in a column, and the conclusionbelow a bar, where  means “therefore”.

CMSC 203 - Discrete Structures

• The general form of a rule of inference is:

• p1

• p2

• .

• .

• .

• pn

• ____

•  q

The rule states that if p1and p2and … and pn are all true, then q is true as well.

These rules of inference can be used in any mathematical argument and do not require any proof.

CMSC 203 - Discrete Structures

q

p  q

_____

  p

Modus tollens

• p

• _____

•  pq

p  q

q  r

_____

 p r

pq

_____

 p

Hypothetical syllogism

Simplification

p

q

_____

 pq

pq

 p

_____

 q

Conjunction

Disjunctive syllogism

CMSC 203 - Discrete Structures

• Just like a rule of inference, an argument consists of one or more hypotheses (or premises) and a conclusion.

• We say that an argument is valid, if whenever all its hypotheses are true, its conclusion is also true.

• However, if any hypothesis is false, even a valid argument can lead to an incorrect conclusion.

• Proof: show that hypotheses  conclusion is true using rules of inference

CMSC 203 - Discrete Structures

• Example:

• “If 101 is divisible by 3, then 1012 is divisible by 9. 101 is divisible by 3. Consequently, 1012 is divisible by 9.”

• Although the argument is valid, its conclusion is incorrect, because one of the hypotheses is false (“101 is divisible by 3.”).

• If in the above argument we replace 101 with 102, we could correctly conclude that 1022 is divisible by 9.

CMSC 203 - Discrete Structures

• Which rule of inference was used in the last argument?

• p: “101 is divisible by 3.”

• q: “1012 is divisible by 9.”

p

p  q

_____

 q

Modus ponens

Unfortunately, one of the hypotheses (p) is false.

Therefore, the conclusion q is incorrect.

CMSC 203 - Discrete Structures

• Another example:

• “If it rains today, then we will not have a barbeque today. If we do not have a barbeque today, then we will have a barbeque tomorrow.Therefore, if it rains today, then we will have a barbeque tomorrow.”

• This is a valid argument: If its hypotheses are true, then its conclusion is also true.

CMSC 203 - Discrete Structures

• Let us formalize the previous argument:

• p: “It is raining today.”

• q: “We will not have a barbecue today.”

• r: “We will have a barbecue tomorrow.”

• So the argument is of the following form:

p  q

q  r

______

 P  r

Hypothetical syllogism

CMSC 203 - Discrete Structures

• Another example:

• Gary is either intelligent or a good actor.

• If Gary is intelligent, then he can count from 1 to 10.

• Gary can only count from 1 to 3.

• Therefore, Gary is a good actor.

• i: “Gary is intelligent.”

• a: “Gary is a good actor.”

• c: “Gary can count from 1 to 10.”

CMSC 203 - Discrete Structures

• i: “Gary is intelligent.”a: “Gary is a good actor.”c: “Gary can count from 1 to 10.”

• Step 1:  c Hypothesis

• Step 2: i  c Hypothesis

• Step 3:  i Modus tollens Steps 1 & 2

• Step 4: a  i Hypothesis

• Step 5: a Disjunctive SyllogismSteps 3 & 4

• Conclusion: a (“Gary is a good actor.”)

CMSC 203 - Discrete Structures

• Yet another example:

• If you listen to me, you will pass CS 320.

• You passed CS 320.

• Therefore, you have listened to me.

• Is this argument valid?

• No, it assumes ((p  q) q)  p.

• This statement is not a tautology. It is false if p is false and q is true.

CMSC 203 - Discrete Structures

Universal instantiation

• x P(x)

• __________

•  P(c) if cU

P(c) for an arbitrary cU

___________________

 x P(x)

Universal generalization

x P(x)

______________________

 P(c) for some element cU

Existential instantiation

P(c) for some element cU

____________________

 x P(x)

Existential generalization

CMSC 203 - Discrete Structures

• Example:

• Every UMB student is a genius.

• George is a UMB student.

• Therefore, George is a genius.

• U(x): “x is a UMB student.”

• G(x): “x is a genius.”

CMSC 203 - Discrete Structures

__________

 P(c) if cU

Universal instantiation

Rules of Inference for Quantified Statements

• The following steps are used in the argument:

• Step 1: x (U(x)  G(x)) Hypothesis

• Step 2: U(George)  G(George) Univ. instantiation using Step 1

Step 3: U(George) Hypothesis

Step 4: G(George) Modus ponens using Steps 2 & 3

CMSC 203 - Discrete Structures

• Direct proof:

• An implication p  q can be proved by showing that if p is true, then q is also true.

• Example: Give a direct proof of the theorem “If n is odd, then n2 is odd.”

• Idea: Assume that the hypothesis of this implication is true (n is odd). Then use rules of inference and known theorems of math to show that q must also be true (n2 is odd).

CMSC 203 - Discrete Structures

• n is odd.

• Then n = 2k + 1, where k is an integer.

• Consequently, n2 = (2k + 1)2.

• = 4k2 + 4k + 1

• = 2(2k2 + 2k) + 1

• Since n2 can be written in this form, it is odd.

CMSC 203 - Discrete Structures

• Indirect proof:

• An implication p  q is equivalent to its contra-positive  q   p. Therefore, we can prove p  q by showing that whenever q is false, then p is also false.

• Example: Give an indirect proof of the theorem “If 3n + 2 is odd, then n is odd.”

• Idea: Assume that the conclusion of this implication is false (n is even). Then use rules of inference and known theorems to show that p must also be false (3n + 2 is even).

CMSC 203 - Discrete Structures

• n is even.

• Then n = 2k, where k is an integer.

• It follows that 3n + 2 = 3(2k) + 2

• = 6k + 2

• = 2(3k + 1)

• Therefore, 3n + 2 is even.

• We have shown that the contrapositive of the implication is true, so the implication itself is also true (If 2n + 3 is odd, then n is odd).

CMSC 203 - Discrete Structures

• Terminology (axiom, theorem, conjecture, etc.)

• Rules of inference (Tables 1 and 2)

• Valid argument (hypotheses and conclusion)

• Construction of valid argument using rules of inference

• Direct and indirect proofs

• Other proof methods (e.g., induction, pigeon hole) will be introduced in later chapters

CMSC 203 - Discrete Structures

and now for something completely different…

• Set Theory

Actually, you will see that logic and set theory are very closely related.

CMSC 203 - Discrete Structures

• Set: Collection of objects (“elements”)

• aA “a is an element of A” “a is a member of A”

• aA “a is not an element of A”

• A = {a1, a2, …, an} “A contains a1, …, an”

• Order of elements is insignificant

• It does not matter how often the same element is listed.

CMSC 203 - Discrete Structures

• Sets A and B are equal if and only if they contain exactly the same elements.

• Examples:

• A = {9, 2, 7, -3}, B = {7, 9, -3, 2} :

A = B

• A = {dog, cat, horse}, B = {cat, horse, squirrel, dog} :

A  B

• A = {dog, cat, horse}, B = {cat, horse, dog, dog} :

A = B

CMSC 203 - Discrete Structures

• “Standard” Sets:

• Natural numbers N = {0, 1, 2, 3, …}

• Integers Z = {…, -2, -1, 0, 1, 2, …}

• Positive Integers Z+ = {1, 2, 3, 4, …}

• Real Numbers R = {47.3, -12, , …}

• Rational Numbers Q = {1.5, 2.6, -3.8, 15, …}

• (correct definitions will follow)

CMSC 203 - Discrete Structures

• A = “empty set/null set”

• A = {z} Note: zA, but z  {z}

• A = {{b, c}, {c, x, d}}

• A = {{x, y}} Note: {x, y} A, but {x, y}  {{x, y}}

• A = {x | P(x)}“set of all x such that P(x)”

• A = {x | x N x > 7} = {8, 9, 10, …}“set builder notation”

CMSC 203 - Discrete Structures

• We are now able to define the set of rational numbers Q:

• Q = {a/b | aZ bZ+}, or

• Q = {a/b | aZ bZ  b0}

• And how about the set of real numbers R?

• R = {r | r is a real number}That is the best we can do. It can neither be defined by enumeration or builder function.

CMSC 203 - Discrete Structures

• A  B “A is a subset of B”

• A  B if and only if every element of A is also an element of B.

• We can completely formalize this:

• A  B  x (xA  xB)

• Examples:

A = {3, 9}, B = {5, 9, 1, 3}, A  B ?

true

A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A  B ?

true

A = {1, 2, 3}, B = {2, 3, 4}, A  B ?

false

CMSC 203 - Discrete Structures

B

C

A

Subsets

• Useful rules:

• A = B  (A  B)  (B  A)

• (A  B)  (B  C)  A  C (see Venn Diagram)

CMSC 203 - Discrete Structures

• Useful rules:

•   A for any set A

• A  A for any set A

• Proper subsets:

• A  B “A is a proper subset of B”

• A  B  x (xA  xB)  x (xB  xA)

• or

• A  B  x (xA  xB)  x (xB  xA)

CMSC 203 - Discrete Structures

• If a set S contains n distinct elements, nN,we call S a finite set with cardinality n.

• Examples:

• A = {Mercedes, BMW, Porsche}, |A| = 3

B = {1, {2, 3}, {4, 5}, 6}

|B| = 4

C = 

|C| = 0

D = { xN | x  7000 }

|D| = 7001

E = { xN | x  7000 }

E is infinite!

CMSC 203 - Discrete Structures

• P(A) “power set of A” (also written as 2A)

• P(A) = {B | B  A} (contains all subsets of A)

• Examples:

• A = {x, y, z}

• P(A)= {, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}

• A = 

• P(A) = {}

• Note: |A| = 0, |P(A)| = 1

CMSC 203 - Discrete Structures

• 1

• 2

• 3

• 4

• 5

• 6

• 7

• 8

• x

• x

• x

• x

• x

• x

• x

• x

• x

• y

• y

• y

• y

• y

• y

• y

• y

• y

• z

• z

• z

• z

• z

• z

• z

• z

• z

The Power Set

• Cardinality of power sets: | P(A) | = 2|A|

• Imagine each element in A has an “on/off” switch

• Each possible switch configuration in A corresponds to one subset of A, thus one element in P(A)

• For 3 elements in A, there are 222 = 8 elements in P(A)

CMSC 203 - Discrete Structures

• The ordered n-tuple (a1, a2, a3, …, an) is an ordered collection of n objects.

• Two ordered n-tuples (a1, a2, a3, …, an) and (b1, b2, b3, …, bn) are equal if and only if they contain exactly the same elements in the same order, i.e. ai = bi for 1  i  n.

• The Cartesian product of two sets is defined as:

• AB = {(a, b) | aA  bB}

CMSC 203 - Discrete Structures

(good, student),

(good, prof),

BA = {

(student, good),

(prof, good),

Cartesian Product

• Example:

• A = {good, bad}, B = {student, prof}

• AB = {

Example: A = {x, y}, B = {a, b, c}AB = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c)}

CMSC 203 - Discrete Structures

• Note that:

• A = 

• A = 

• For non-empty sets A and B: AB  AB  BA

• |AB| = |A||B|

• The Cartesian product of two or more sets is defined as:

• A1A2…An = {(a1, a2, …, an) | aiA for 1  i  n}

CMSC 203 - Discrete Structures

• Union: AB = {x | xA  xB}

• Example: A = {a, b}, B = {b, c, d}

• AB = {a, b, c, d}

• Intersection: AB = {x | xA  xB}

• Example: A = {a, b}, B = {b, c, d}

• AB = {b}

CMSC 203 - Discrete Structures

• Two sets are called disjoint if their intersection is empty, that is, they share no elements:

• AB = 

• The difference between two sets A and B contains exactly those elements of A that are not in B:

• A-B = {x | xA  xB}Example: A = {a, b}, B = {b, c, d}, A-B = {a}

CMSC 203 - Discrete Structures

• The complement of a set A contains exactly those elements under consideration that are not in A:

• Ac = U-A

• Example: U = N, B = {250, 251, 252, …}

• Bc = {0, 1, 2, …, 248, 249}

CMSC 203 - Discrete Structures

• Table 1 in Section 1.7 shows many useful equations

• How can we prove A(BC) = (AB)(AC)?

• Method I:logical equivalent

• xA(BC)

• xA  x(BC)

• xA  (xB  xC)

• (xA  xB)  (xA  xC) (distributive law)

• x(AB)  x(AC)

• x(AB)(AC)

Every logical expression can be transformed into an equivalent expression in set theory and vice versa.

CMSC 203 - Discrete Structures

• BC

• A(BC)

• AB

• AC

• (AB) (AC)

• 0 0 0

• 0

• 0

• 0

• 0

• 0

• 0 0 1

• 0

• 0

• 0

• 1

• 0

• 0 1 0

• 0

• 0

• 1

• 0

• 0

• 0 1 1

• 1

• 1

• 1

• 1

• 1

• 1 0 0

• 0

• 1

• 1

• 1

• 1

• 1 0 1

• 0

• 1

• 1

• 1

• 1

• 1 1 0

• 0

• 1

• 1

• 1

• 1

• 1 1 1

• 1

• 1

• 1

• 1

• 1

Set Operations

• Method II: Membership table

• 1 means “x is an element of this set”0 means “x is not an element of this set”

CMSC 203 - Discrete Structures

• Functions

CMSC 203 - Discrete Structures

• A function f from a set A to a set B is an assignment of exactly one element of B to each element of A.

• We write

• f(a) = b

• if b is the unique element of B assigned by the function f to the element a of A.

• If f is a function from A to B, we write

• f: AB

• (note: Here, ““ has nothing to do with if… then)

CMSC 203 - Discrete Structures

• If f:AB, we say that A is the domain of f and B is the codomain of f.

• If f(a) = b, we say that b is the image of a and a is the pre-image of b.

• The range of f:AB is the set of all images of elements of A.

• We say that f:AB maps A to B.

CMSC 203 - Discrete Structures

• Let us take a look at the function f:PC with

• P = {Linda, Max, Kathy, Peter}

• C = {Boston, New York, Hong Kong, Moscow}

• f(Linda) = Moscow

• f(Max) = Boston

• f(Kathy) = Hong Kong

• f(Peter) = New York

• Here, the range of f is C.

CMSC 203 - Discrete Structures

• Let us re-specify f as follows:

• f(Linda) = Moscow

• f(Max) = Boston

• f(Kathy) = Hong Kong

• f(Peter) = Boston

• Is f still a function?

yes

What is its range?

{Moscow, Boston, Hong Kong}

CMSC 203 - Discrete Structures

• f(x)

Linda

Boston

• Linda

• Moscow

Max

New York

• Max

• Boston

Kathy

Hong Kong

• Kathy

• Hong Kong

Peter

Moscow

• Peter

• Boston

Functions

• Other ways to represent f:

CMSC 203 - Discrete Structures

• If the domain of our function f is large, it is convenient to specify f with a formula, e.g.:

• f:RR

• f(x) = 2x

• f(1) = 2

• f(3) = 6

• f(-3) = -6

CMSC 203 - Discrete Structures

• Let f1 and f2 be functions from A to R.

• Then the sum and the product of f1 and f2 are also functions from A to R defined by:

• (f1 + f2)(x) = f1(x) + f2(x)

• (f1f2)(x) = f1(x) f2(x)

• Example:

• f1(x) = 3x, f2(x) = x + 5

• (f1 + f2)(x) = f1(x) + f2(x) = 3x + x + 5 = 4x + 5

• (f1f2)(x) = f1(x) f2(x) = 3x (x + 5) = 3x2 + 15x

CMSC 203 - Discrete Structures

• We already know that the range of a function f:AB is the set of all images of elements aA.

• If we only regard a subset SA, the set of all images of elements sS is called the image of S.

• We denote the image of S by f(S):

• f(S) = {f(s) | sS}

CMSC 203 - Discrete Structures

• Let us look at the following well-known function:

• f(Linda) = Moscow

• f(Max) = Boston

• f(Kathy) = Hong Kong

• f(Peter) = Boston

• What is the image of S = {Linda, Max} ?

• f(S) = {Moscow, Boston}

• What is the image of S = {Max, Peter} ?

• f(S) = {Boston}

CMSC 203 - Discrete Structures

• A function f:AB is said to be one-to-one (or injective), if and only if

• x, yA (f(x) = f(y)  x = y)

• In other words: f is one-to-one if and only if it does not map two distinct elements of A onto the same element of B.

CMSC 203 - Discrete Structures

g(Linda) = Moscow

g(Max) = Boston

g(Kathy) = Hong Kong

g(Peter) = New York

Is g one-to-one?

Yes, each element is assigned a unique element of the image.

• And again…

• f(Linda) = Moscow

• f(Max) = Boston

• f(Kathy) = Hong Kong

• f(Peter) = Boston

• Is f one-to-one?

• No, Max and Peter are mapped onto the same element of the image.

CMSC 203 - Discrete Structures

• How can we prove that a function f is one-to-one?

• Whenever you want to prove something, first take a look at the relevant definition(s):

• x, yA (f(x) = f(y)  x = y)

• Example:

• f:RR

• f(x) = x2

• Disproof by counterexample:

• f(3) = f(-3), but 3  -3, so f is not one-to-one.

CMSC 203 - Discrete Structures

• … and yet another example:

• f:RR

• f(x) = 3x

• One-to-one: x, yA (f(x) = f(y)  x = y)

• To show: f(x)  f(y) whenever x  y

• x  y

• 3x  3y

• f(x)  f(y),

so if x  y, then f(x)  f(y), that is, f is one-to-one.

CMSC 203 - Discrete Structures

• A function f:AB with A,B  R is called strictly increasing, if

• x,yA (x < y  f(x) < f(y)),

• and strictly decreasing, if

• x,yA (x < y  f(x) > f(y)).

• Obviously, a function that is either strictly increasing or strictly decreasing is one-to-one.

CMSC 203 - Discrete Structures

• A function f:AB is called onto, or surjective, if and only if for every element bB there is an element aA with f(a) = b.

• In other words, f is onto if and only if its range is its entire codomain.

• A function f: AB is a one-to-one correspondence, or a bijection, if and only if it is both one-to-one and onto.

• Obviously, if f is a bijection and A and B are finite sets, then |A| = |B|.

CMSC 203 - Discrete Structures

• Examples:

• In the following examples, we use the arrow representation to illustrate functions f:AB.

• In each example, the complete sets A and B are shown.

CMSC 203 - Discrete Structures

Boston

Max

New York

Kathy

Hong Kong

Peter

Moscow

Properties of Functions

• Is f injective?

• No.

• Is f surjective?

• No.

• Is f bijective?

• No.

CMSC 203 - Discrete Structures

Boston

Max

New York

Kathy

Hong Kong

Peter

Moscow

Properties of Functions

• Is f injective?

• No.

• Is f surjective?

• Yes.

• Is f bijective?

• No.

Paul

CMSC 203 - Discrete Structures

Boston

Max

New York

Kathy

Hong Kong

Peter

Moscow

Lübeck

Properties of Functions

• Is f injective?

• Yes.

• Is f surjective?

• No.

• Is f bijective?

• No.

CMSC 203 - Discrete Structures

Boston

Max

New York

Kathy

Hong Kong

Peter

Moscow

Lübeck

Properties of Functions

• Is f injective?

• No! f is not evena function!

CMSC 203 - Discrete Structures

Linda

Boston

• Is f injective?

• Yes.

• Is f surjective?

• Yes.

• Is f bijective?

• Yes.

Max

New York

Kathy

Hong Kong

Peter

Moscow

Helena

Lübeck

CMSC 203 - Discrete Structures

• An interesting property of bijections is that they have an inverse function.

• The inverse function of the bijection f:AB is the function f-1:BA with

• f-1(b) = a whenever f(a) = b.

CMSC 203 - Discrete Structures

Example:

f(Linda) = Moscow

f(Max) = Boston

f(Kathy) = Hong Kong

f(Peter) = Lübeck

f(Helena) = New York

Clearly, f is bijective.

The inverse function f-1 is given by:

f-1(Moscow) = Linda

f-1(Boston) = Max

f-1(Hong Kong) = Kathy

f-1(Lübeck) = Peter

f-1(New York) = Helena

Inversion is only possible for bijections(= invertible functions)

CMSC 203 - Discrete Structures

f-1

Inversion

Linda

Boston

• f-1:CP is no function, because it is not defined for all elements of C and assigns two images to the pre-image New York.

Max

New York

Kathy

Hong Kong

Peter

Moscow

Helena

Lübeck

CMSC 203 - Discrete Structures

• The composition of two functions g:AB and f:BC, denoted by fg, is defined by

• (fg)(a) = f(g(a))

• This means that

• first, function g is applied to element aA, mapping it onto an element of B,

• then, function f is applied to this element of B, mapping it onto an element of C.

• Therefore, the composite function maps from A to C.

CMSC 203 - Discrete Structures

• Example:

• f(x) = 7x – 4, g(x) = 3x,

• f:RR, g:RR

• (fg)(5) = f(g(5)) = f(15) = 105 – 4 = 101

• (fg)(x) = f(g(x)) = f(3x) = 21x - 4

CMSC 203 - Discrete Structures

• Composition of a function and its inverse:

• (f-1f)(x) = f-1(f(x)) = x

• The composition of a function and its inverse is the identity function i(x) = x.

CMSC 203 - Discrete Structures

• Thegraphof a functionf:AB is the set of ordered pairs {(a, b) | aA and f(a) = b}.

• The graph is a subset of AB that can be used to visualize f in a two-dimensional coordinate system.

CMSC 203 - Discrete Structures

• The floor and ceiling functions map the real numbers onto the integers (RZ).

• The floor function assigns to rR the largest zZ with z  r, denoted by r.

• Examples: 2.3 = 2, 2 = 2, 0.5 = 0, -3.5 = -4

• The ceiling function assigns to rR the smallest zZ with z  r, denoted by r.

• Examples: 2.3 = 3, 2 = 2, 0.5 = 1, -3.5 = -3

CMSC 203 - Discrete Structures

• I recommend Exercises 1 and 15 in Section 1.6.

• It may also be useful to study the graph displays in that section.

• Another question: What do all graph displays for any function f:RR have in common?

CMSC 203 - Discrete Structures