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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

- 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

The Statement/Proposition Game

- “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

The Statement/Proposition Game

- “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

The Statement/Proposition Game

- “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

The Statement/Proposition Game

- “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

The Statement/Proposition Game

- “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

The Statement/Proposition Game

- “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

The Statement/Proposition Game

- “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

Combining Propositions

- 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

Logical Operators (Connectives)

- 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

Statements and Operators

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

CMSC 203 - Discrete Structures

Statements and Operations

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

CMSC 203 - Discrete Structures

Exercises

- 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

Equivalent Statements

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

CMSC 203 - Discrete Structures

Tautologies and Contradictions

- A tautology is a statement that is always true.
- Examples:
- R (R)
- (PQ) (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

Equivalence

- 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

- 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, (PQ) ( P) ( Q)
- Law with implication P Q P Q

CMSC 203 - Discrete Structures

Exercises

- 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

Summary, Sections 1.1, 1.2

- 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 Functions & Predicates

- 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

Propositional Functions

- 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

Universal Quantification

- 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

Universal Quantification

- 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

Existential Quantification

- 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

Existential Quantification

- 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

Quantification

- Another example:
- Let the universe of discourse be the real numbers.
- What does xy (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

Disproof by Counterexample

- 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

Negation

- (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

Nested Quantifier

- 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

Summary, Sections 1.3, 1.4

- 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

- 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

Terminology

- 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

Terminology

- 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

- 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

Rules of Inference

- 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

Rules of Inference

q

p q

_____

p

Modus tollens

Addition

- p
- _____
- pq

p q

q r

_____

p r

pq

_____

p

Hypothetical syllogism

Simplification

p

q

_____

pq

pq

p

_____

q

Conjunction

Disjunctive syllogism

CMSC 203 - Discrete Structures

Arguments

- 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

Arguments

- 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

Arguments

- 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

Arguments

- 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

Arguments

- 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

Arguments

- 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

Arguments

- 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

Arguments

- 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

Rules of Inference for Quantified Statements

Universal instantiation

- x P(x)
- __________
- P(c) if cU

P(c) for an arbitrary cU

___________________

x P(x)

Universal generalization

x P(x)

______________________

P(c) for some element cU

Existential instantiation

P(c) for some element cU

____________________

x P(x)

Existential generalization

CMSC 203 - Discrete Structures

Rules of Inference for Quantified Statements

- 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 cU

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

Proving Theorems

- 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

Proving Theorems

- 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

Proving Theorems

- 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

Proving Theorems

- 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

Summary, Section 1.5

- 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 Theory

- Set: Collection of objects (“elements”)
- aA “a is an element of A” “a is a member of A”
- aA “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

Set Equality

- 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

Examples for Sets

- “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

Examples for Sets

- A = “empty set/null set”
- A = {z} Note: zA, 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

Examples for Sets

- We are now able to define the set of rational numbers Q:
- Q = {a/b | aZ bZ+}, or
- Q = {a/b | aZ bZ b0}
- 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

Subsets

- 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 (xA xB)
- 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

Subsets

- 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 (xA xB) x (xB xA)
- or
- A B x (xA xB) x (xB xA)

CMSC 203 - Discrete Structures

Cardinality of Sets

- If a set S contains n distinct elements, nN,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 = { xN | x 7000 }

|D| = 7001

E = { xN | x 7000 }

E is infinite!

CMSC 203 - Discrete Structures

The Power Set

- 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

- 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 222 = 8 elements in P(A)

CMSC 203 - Discrete Structures

Cartesian Product

- 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:
- AB = {(a, b) | aA bB}

CMSC 203 - Discrete Structures

(good, student),

(good, prof),

(bad, student),

(prof, bad)}

BA = {

(student, good),

(prof, good),

(student, bad),

Cartesian Product- Example:
- A = {good, bad}, B = {student, prof}
- AB = {

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

CMSC 203 - Discrete Structures

Cartesian Product

- Note that:
- A =
- A =
- For non-empty sets A and B: AB AB BA
- |AB| = |A||B|
- The Cartesian product of two or more sets is defined as:
- A1A2…An = {(a1, a2, …, an) | aiA for 1 i n}

CMSC 203 - Discrete Structures

Set Operations

- Union: AB = {x | xA xB}
- Example: A = {a, b}, B = {b, c, d}
- AB = {a, b, c, d}
- Intersection: AB = {x | xA xB}
- Example: A = {a, b}, B = {b, c, d}
- AB = {b}

CMSC 203 - Discrete Structures

Set Operations

- Two sets are called disjoint if their intersection is empty, that is, they share no elements:
- AB =
- The difference between two sets A and B contains exactly those elements of A that are not in B:
- A-B = {x | xA xB}Example: A = {a, b}, B = {b, c, d}, A-B = {a}

CMSC 203 - Discrete Structures

Set Operations

- 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

Set Identity

- Table 1 in Section 1.7 shows many useful equations
- How can we prove A(BC) = (AB)(AC)?
- Method I:logical equivalent
- xA(BC)
- xA x(BC)
- xA (xB xC)
- (xA xB) (xA xC) (distributive law)
- x(AB) x(AC)
- x(AB)(AC)
Every logical expression can be transformed into an equivalent expression in set theory and vice versa.

CMSC 203 - Discrete Structures

- BC

- A(BC)

- AB

- AC

- (AB) (AC)

- 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

- 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

- 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: AB
- (note: Here, ““ has nothing to do with if… then)

CMSC 203 - Discrete Structures

Functions

- If f:AB, 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:AB is the set of all images of elements of A.
- We say that f:AB maps A to B.

CMSC 203 - Discrete Structures

Functions

- Let us take a look at the function f:PC 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

Functions

- 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

- Other ways to represent f:

CMSC 203 - Discrete Structures

Functions

- If the domain of our function f is large, it is convenient to specify f with a formula, e.g.:
- f:RR
- f(x) = 2x
- This leads to:
- f(1) = 2
- f(3) = 6
- f(-3) = -6
- …

CMSC 203 - Discrete Structures

Functions

- 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

Functions

- We already know that the range of a function f:AB is the set of all images of elements aA.
- If we only regard a subset SA, the set of all images of elements sS is called the image of S.
- We denote the image of S by f(S):
- f(S) = {f(s) | sS}

CMSC 203 - Discrete Structures

Functions

- 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

Properties of Functions

- A function f:AB is said to be one-to-one (or injective), if and only if
- x, yA (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

Properties of Functions

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

Properties of Functions

- 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, yA (f(x) = f(y) x = y)
- Example:
- f:RR
- f(x) = x2
- Disproof by counterexample:
- f(3) = f(-3), but 3 -3, so f is not one-to-one.

CMSC 203 - Discrete Structures

Properties of Functions

- … and yet another example:
- f:RR
- f(x) = 3x
- One-to-one: x, yA (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

Properties of Functions

- A function f:AB with A,B R is called strictly increasing, if
- x,yA (x < y f(x) < f(y)),
- and strictly decreasing, if
- x,yA (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

Properties of Functions

- A function f:AB is called onto, or surjective, if and only if for every element bB there is an element aA with f(a) = b.
- In other words, f is onto if and only if its range is its entire codomain.
- A function f: AB 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

Properties of Functions

- Examples:
- In the following examples, we use the arrow representation to illustrate functions f:AB.
- 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

Properties of Functions

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

Inversion

- An interesting property of bijections is that they have an inverse function.
- The inverse function of the bijection f:AB is the function f-1:BA with
- f-1(b) = a whenever f(a) = b.

CMSC 203 - Discrete Structures

Inversion

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

InversionLinda

Boston

- f-1:CP 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

Composition

- The composition of two functions g:AB and f:BC, denoted by fg, is defined by
- (fg)(a) = f(g(a))
- This means that
- first, function g is applied to element aA, 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

Composition

- Example:
- f(x) = 7x – 4, g(x) = 3x,
- f:RR, g:RR
- (fg)(5) = f(g(5)) = f(15) = 105 – 4 = 101
- (fg)(x) = f(g(x)) = f(3x) = 21x - 4

CMSC 203 - Discrete Structures

Composition

- Composition of a function and its inverse:
- (f-1f)(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

Graphs

- Thegraphof a functionf:AB is the set of ordered pairs {(a, b) | aA and f(a) = b}.
- The graph is a subset of AB that can be used to visualize f in a two-dimensional coordinate system.

CMSC 203 - Discrete Structures

Floor and Ceiling Functions

- The floor and ceiling functions map the real numbers onto the integers (RZ).
- The floor function assigns to rR the largest zZ with z r, denoted by r.
- Examples: 2.3 = 2, 2 = 2, 0.5 = 0, -3.5 = -4
- The ceiling function assigns to rR the smallest zZ with z r, denoted by r.
- Examples: 2.3 = 3, 2 = 2, 0.5 = 1, -3.5 = -3

CMSC 203 - Discrete Structures

Exercises

- 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:RR have in common?

CMSC 203 - Discrete Structures

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