LOGIC

LOGIC

• The discipline that deals with the methods of

reasoning.

• Logical reasoning is used in mathematics to prove

theorems, in computer science to verify the

correctness of programs and to prove theorems.

PROPOSITIONS

• A proposition is a declarative sentence that is either

true or false, but not both.

• Examples:

“the earth is round”,

“2+3=5”

Compound PropositionsPropositional variables (statements) can be combined together by logical connectives to obtain compound propositions (compound statements).

For example,

if we have two propositions, e.g.

(1) “the sun is shining”,

(2) “it is cold”,

we can combine them together by the connective “and” to form the compound statement

“the sun is shining and it is cold”.

logical operators

• The truth value of an entire compound proposition

depends only on the truth values of the statements

being combined and on the types of connectives

being used.

• These connectives in logic are called logical

operators.

logical operatorsOperator Symbol Usage

Negation not

Conjunction and

Disjunction or

Exclusive or /v xor

Conditional if,then

Biconditional iff

Example 1Let P be the proposition logic is fun and Q be the proposition today is Friday. Express each of the following compound propositions in symbolic form:

• Logic is not fun and today is Friday

• Today is not Friday, nor is logic fun

• Either logic is fun or it is Friday

SOLUTION

(a) (not P) and Q

(b) (not Q) and (not P)

(c) P or Q

truth table.

To determine the truth value of compound proposition

we need to know the effect of logical operators.

These can be summarized using a truth table.

Truth Table -NOT /NEGATION

(¬)

A truth table that expresses a relationship a propositional variable (p) and a

negation of that variable (¬p):

(p) (¬p)

T F

F T

Truth Table for AND/CONJUCTION (λ), OR/DISJUNCTION

(v), EXCLUSIVE OR (v), IMPLICATION (→), and

EQUIVALENCE (↔)

(p) (q) (p Λ q) (p V q) (p V q) (p →q) (p ↔ q)

T T

T F

F T

F F

Truth Table for AND/CONJUCTION (λ), OR/DISJUNCTION

(v), EXCLUSIVE OR (v), IMPLICATION (→), and

EQUIVALENCE (↔)

(p) (q) (p Λ q) (p V q) (p V q) (p →q) (p ↔ q)

T T T T F T T

T F F T T F F

F T F T T T F

F F F F F T T

logically equivalent

In implication/ conditional propositions

(p →q),

the preposition p is sometimes called the antecedent

(hypothesis) and q is called the consequent (conclusion).

Two compound propositions are logically equivalent (≡) if

they have the same truth value for every possible set of

truth values of the constituent propositions.

Example 1

Show that the proposition

(not (P and (not Q)))

is logically equivalent to the proposition

((not P) or Q)

Solution

P Q not P not Q P and (not Q) R S

T T

T F

F T

F F

We construct a combined truth table for the compound propositions R= (not (P and (not Q))) and S= ((not P) or Q). Intermediary columns are used as required to build up each of the expressions from P and Q.

Solution

P Q not P not Q P and (not Q) R S

T T F F F T T

T F F T T F F

F T T F F T T

F F T T F T T

We construct a combined truth table for the compound propositions R= (not (P and (not Q))) and S= ((not P) or Q). Intermediary columns are used as required to build up each of the expressions from P and Q.

Since the last two columns are identical, R is logically equivalent to S (R≡S).

More about conditionals

Given the conditional proposition (p →q), we define

the following:

• The converse of p →q : q → p

• The inverse of p →q : 𝑝 → 𝑝

• The contrapositive of p →q : 𝑞 → 𝑝

The converse of p →q : q → p

The inverse of p →q : 𝑝 → 𝑝

inverse: p →

Example 3:

• Original: If you finish dinner, you can have dessert.

• Converse: If you are eating dessert, you must have

finished dinner.

• Inverse: If you do not finish dinner, you cannot

have dessert.

The contrapositive of p →q : 𝑞 → 𝑝Examples 4:

1. If you want to get there faster, you need to take a plane.

contrapositive:

If you don’t take a plane, you cannot get there faster.

2. If you eat too much turkey, you will feel sleepy.

contrapositive:

If you do not want to feel sleepy, you better not eat that much turkey.

3. If you want to see sunrise, you have to get up early.

contrapositive:

If you do not get up early, you cannot see sunrise.

Truth Table for Converse, Inverse,

and Contrapositive

(p) (q) (p →q) (q → p) ( 𝒑 → 𝒒 ) ( 𝒒 → 𝒑)

T T

T F

F T

F F

Truth Table for Converse, Inverse,

and Contrapositive

(p) (q) (p →q) (q → p) ( 𝒑 → 𝒒 ) ( 𝒒 → 𝒑)

T T T T T T

T F F T T F

F T T F F T

F F T T T T

Tautology, Contradiction and

Contingency

• Tautology: A compound proposition that is always true, no matter what the truth values of proposition that occur

• Contradiction: A compound proposition that is always false, no matter what the truth values of proposition that occur

• Contingency: A compound proposition that is neither a tautology nor contradictory.

Example 5

• Show that 𝑝 V 𝑝 is a tautology

SOLUTION

Constructing the truth table for p V 𝑝 , we have :

Note that p V 𝑝 is always true (no matter what proposition is substituted for

p) and is therefore a tautology.

(p) ( 𝒑) (p V 𝒑)

T F T

F T T

PREDICATES AND

QUANTIFIES

A predicates describes a property of one or several objects or individuals that we called variables which is either true or false depending on the values assigned to the variables.

Example 4

x is an integer satisfying x=x2

is a predicate since it is true for x=0 or 1

and false for all other values.

quantifiers

Logical operations can now be applied to predicates. In general, the truth of a compound predicate ultimately depends on the values assigned to the variables involved. However, there are further logical operators, known as quantifiers, which when applied to a predicate produce either a true proposition. The following are the two predicate quantifiers,

Universal quantifier: The symbol is ∀

• ∀ x is read as ‘for all x’ or ‘for every x’

Existential quantifier: The symbol is ∃

• ∃ x is read as ‘there exist at least one x’ or ‘for some x’

Example 6

Which of the following statements are true, and which are

false?

1. All triangles have the sum of their angles equal to 180⁰.

2. All cats have tails.

3. There is an integer x satisfying x2 = 2.

4. There is a prime number which is not odd

SOLUTION

1. True

2. False. Many cats have no tails

3. False

4. True. The number 2 is both a prime number and it

is even.

SET

Set Theoretic Operations

Set theoretic operations allow us to build new sets out of old,

just as the logical connectives allowed us to create compound

propositions from simpler propositions.

Given sets A and B, the set theoretic operators are:

• Union ()

• Intersection ()

• Difference (-)

• Complement (“—”)

• Symmetric Difference ()

give us new sets AB, AB, A-B, AB, andA .

Sets

• Set-builder method

• A = { x | x S, P(x) } or A = { x S | P(x) }

A is the set of all elements x of S, such that x satisfies the property P

Example 1:

If X = {2,4,6,8,10}, then in set-builder notation, X can be described

as

X = {n Z | n is even and 2 n 10}

Sets• Standard Symbols which denote sets of numbers

• N : The set of all natural numbers (i.e.,all positive integers)

• Z : The set of all integers

• Z* : The set of all nonzero integers

• E : The set of all even integers

• Q : The set of all rational numbers

• Q* : The set of all nonzero rational numbers

• Q+ : The set of all positive rational numbers

• R : The set of all real numbers

• R* : The set of all nonzero real numbers

• R+ : The set of all positive real numbers

• C : The set of all complex numbers

• C* : The set of all nonzero complex numbers

Sets

• Subsets

• “X is a subset of Y” is written as X Y

• “X is not a subset of Y” is written as X Y

Example 2:

• X = {a,e,i,o,u}, Y = {a, i, u} and z = {b,c,d,f,g}

• Y X, since every element of Y is an element of X

• Y Z, since a Y, but a Z

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Sets

• Superset

X and Y are sets. If X Y, then “X is contained in Y” or “Y contains X” or Y is a superset of X, written Y X

• Proper Subset

X and Y are sets. X is a proper subset of Y

if X Y and there exists at least one element in Y that is not in X. This is written X Y.

Example 3:

X = {a,e,i,o,u}, Y = {a,e,i,o,u,y}

X Y , since y Y, but y X

Sets • Set Equality

X and Y are sets. They are said to be equal if every element of X is an element of Y and every element of Y is an element of X, i.e. X Y and Y X

Example 4:

• {1,2,3} = {2,3,1}

• X = {red, blue, yellow} and Y = {c | c is a primary color} Therefore, X=Y

• Empty (Null) Set

• A Set is Empty (Null) if it contains no elements.

• The Empty Set is written as

• The Empty Set is a subset of every set

Sets

• Finite and Infinite Sets

• X is a set. If there exists a nonnegative integer n such that X

has n elements, then X is called a finite set with n elements.

• If a set is not finite, then it is an infinite set.

Example 5:

• Y = {1,2,3} is a finite set

• P = {red, blue, yellow} is a finite set

• E , the set of all even integers, is an infinite set

• , the Empty Set, is a finite set with 0 elements

Sets• Cardinality of Sets

• Let S be a finite set with n distinct elements, where n ≥ 0. Then |S| = n , where the cardinality (number of elements) of S is n

Example 6:

• If P = {red, blue, yellow}, then |P| = 3

• Singleton

• A set with only one element is a singleton

• Example:

• H = { 4 }, |H| = 1, H is a singleton

Sets

• Power Set

For any set X ,the power set of X ,written P(X),is the set of all

subsets of X

Example 7:

• If X = {red, blue, yellow}, then P(X) = { , {red}, {blue},

{yellow}, {red,blue}, {red, yellow}, {blue, yellow}, {red, blue,

yellow} }

• Universal Set

An arbitrarily chosen, but fixed set

Venn Diagrams

• Venn diagrams are useful in representing sets and set operations.

• Various sets are represented by circles inside a big rectangle representing the universe of reference.

• Common universal sets

• R =real numbers

• N =natural numbers ={0,1,2,3,…}

• Z =all integers

• Z+ = is the set of positive integers

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UNION

AB = { x | x A x B }Elements in at least one of the two sets:

A B

U

AB

Example 8:

If A = {1,2,3,4,5} and B = {5,6,7,8,9}, then A∪B = {1,2,3,4,5,6,7,8,9}

Intersection

AB = { x | x A x B }

Elements in exactly one of the two sets:

A B

U

AB

Example 9:

If A = {1,2,3,4,5} and B = {5,6,7,8,9}, then A ∩ B = {5}

Disjoint SetsDEF: If A and B have no common elements, they

are said to be disjoint, i.e. A B = ={ } .

A B

U

Example 10:

If A = {1,2,3,4,} and B = {6,7,8,9}, then A ∩ B=

Disjoint UnionWhen A and B are disjoint, the disjoint union operationis well defined. The circle above the union symbol indicates disjointedness.

A B

U

BA

Disjoint Union

FACT: In a disjoint union of finite sets, cardinality of

the union is the sum of the cardinalities. I.e.

BABA

Example 11 :

A={a,b,c} |A|= 3

B={0.2, 1, 3, 4.5, 3, 9} |B|= ?

Set Difference

A-B = { x | x A x B }

Elements in first set but not second:

A

B

U

A-B

Example 12:

If A = {a,b,c,d} and B = {c,d,e,f}, then A – B = {a,b}

and B – A = {e,f}

Symmetric Difference

A B

UAB

AB = { x | x A x B }

Elements in exactly one of the two sets:

Complement

A = { x | x A }

Elements not in the set (unary operator):

A

U

A

Example 13:

If U = {a,b,c,d,e,f} and A= {c,d,e,f}, then B’ = {a,b}

Set Identities

Table 1, Rosen p. 49

• Identity laws

• Domination laws

• Idempotent laws

• Double complementation

• Commutativity

• Associativity

• Distribuitivity

• DeMorgan

This table is gotten from the previous table of logical identities by rewriting as follows:

• disjunction “” becomes union “”

• conjunction “” becomes intersection “”

• negation “” becomes complementation “–”

• false “F” becomes the empty set

• true “T” becomes the universe of reference U

L5 46

Sets• Ordered Pair

• X and Y are sets. If x X and y Y, then an ordered pair is written (x,y)

• Order of elements is important. (x,y) is not necessarily equal to (y,x)

• Cartesian Product

• The Cartesian product of two sets X and Y ,written X × Y ,is the set

• X × Y ={(x,y)|x ∈ X , y ∈ Y}

• For any set X, X × = = × X

Example 14:

• X = {a,b}, Y = {c,d}

• X × Y = {(a,c), (a,d), (b,c), (b,d)}

• Y × X = {(c,a), (d,a), (c,b), (d,b)}

Sets

• Diagonal of a Set

• For a set X ,the set δx , is the diagonal of X, defined by

δx = {(x,x) | x ∈ X}

Example 15:

• X = {a,b,c}, δx = {(a,a), (b,b), (c,c)}

FUNCTION

Definition of Functions

• Given any sets A, B, a function f from (or “mapping”)

A to B (f:AB) is an assignment of exactly one

element f(x)B

to each element xA.

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Graphical Representations• Functions can be represented graphically in several ways:

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B

• •

A

a b

f

f

•

•••

•

•

•

•

•x

y

PlotGraph

Like Venn diagrams

A B

Some Function Terminology

• If f:AB, and f(a)=b (where aA & bB), then:

• A is the domain of f.

• B is the codomain of f.

• b is the image of a under f.

• a is a pre-image of b under f.

• In general, b may have more than one pre-image.

• The range RB of f is {b | a f(a)=b }.

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• •

A

a b

f

f

B

Range vs. Codomain - Example• Suppose that: “f is a function mapping students in this class to the set

of grades {A,B,C,D,E}.”

• At this point, you know f’s codomain is: __________, and its

range is ________.

• Suppose the grades turn out all As and Bs.

• Then the range of f is _________, but its codomain is

__________________.

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{A,B,C,D,E}

unknown!

{A,B}

still {A,B,C,D,E}!

Function Addition/Multiplication

• We can add and multiply functions

f,g:RR:

• (f g):RR, where (f g)(x) = f(x) g(x)

• (f × g):RR, where (f × g)(x) = f(x)× g(x)

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Function Composition• For functions g:AB and f:BC, there is a special operator called compose

(“°”).

• It composes (i.e., creates) a new function out of f,g by applying f to

the result of g.

(f °g):AC, where (f ° g)(a) = f(g(a)).

• Note g(a)B, so f(g(a)) is defined and C.

• The range of g must be a subset of f’s domain!!

• Note that ° (like Cartesian , but unlike +,,) is non-commuting.

(In general, f ° g g ° f.)

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Function Composition

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Images of Sets under Functions

• Given f:AB, and SA,

• The image of S under f is simply the set of all images

(under f) of the elements of S.

f(S) : {f(s) | sS}

: {b | sS: f(s)=b}.

• Note the range of f can be defined as simply the

image (under f) of f’s domain!

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One-to-One Functions• A function is one-to-one (1-1), or injective, or an injection, iff every

element of its range has only one pre-image.

• Only one element of the domain is mapped to any given one

element of the range.

• Domain & range have same cardinality. What about codomain?

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One-to-One Illustration• Graph representations of functions that are (or not) one-to-

one:

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•

•••

•

•

•

•

•

One-to-one

••••

••

•

•

•

Not one-to-one

••••

••

•

•

•Not even a

function!

Sufficient Conditions for 1-1ness• Definitions (for functions f over numbers):

• f is strictly (or monotonically) increasing iff x>y f(x)>f(y) for all x,y in

domain;

• f is strictly (or monotonically) decreasing iff x>y f(x)<f(y) for all x,y in

domain;

• If f is either strictly increasing or strictly decreasing, then f is

one-to-one.

• Example . f(x)=x3

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Onto (Surjective) Functions

• A function f:AB is onto or surjective or a surjection iff its

range is equal to its codomain (bB, aA: f(a)=b).

• An onto function maps the set A onto (over, covering) the

entirety of the set B, not just over a piece of it.

• Example: for domain & codomain R, x3 is onto, whereas x2 isn’t.

(Why not?)

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Illustration of Onto• Some functions that are or are not onto their codomains:

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Onto

(but not 1-1)

•

•••

•

•

•

•

•

Not Onto

(or 1-1)

•

•••

•

•

•

•

•

Both 1-1

and onto

•

•••

•

•

•

•

1-1 but

not onto

•

•••

•

•

•

•

•

Bijections

• A function f is a one-to-one correspondence, or a bijection,

or reversible, or invertible, iff it is both one-to-one and

onto.

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Inverse of a Function

• For bijections f:AB, there exists an inverse of f,

written f -1:BA, which is the unique function such

that:

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Iff - 1

Inverse of a function (cont’d)

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The Identity Function

• For any domain A, the identity function I:AA

(variously written, IA, 1, 1A) is the unique function

such that aA: I(a)=a.

• Some identity functions you’ve seen:

• ing with T, ing with F, ing with , ing with U.

• Note that the identity function is both one-to-one

and onto (bijective).

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Graphs of Functions

• We can represent a function f:AB as a set of ordered pairs

{(a,f(a)) | aA}.

• Note that a, there is only one pair (a, f(a)).

• For functions over numbers, we can represent an ordered pair

(x,y) as a point on a plane. A function is then drawn as a

curve (set of points) with only one y for each x.

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