Logical equivalence
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Logical equivalence. Two propositions are said to be logically equivalent if their truth tables are identical. Example: ~p  q is logically equivalent to p  q. T. T. T. T. F. F. T. F. F. T. T. T. F. F. T. T. Converse (1).

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Logical equivalence

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Logical equivalence

Logical equivalence

  • Two propositions are said to be logically

    equivalent if their truth tables are identical.

  • Example: ~p  q is logically equivalent to p  q

T

T

T

T

F

F

T

F

F

T

T

T

F

F

T

T

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Converse 1

Converse (1)

  • The converse of a conditional statement is formed by interchanging the hypothesis and conclusion of the original statement.In other words, the parts of the sentence change places but the words "if" and "then" do not leave their places.

  • Conditional:  "If 9 is an odd number, then 9 is divisible by 2.“

  • Converse: "If 9 is divisible by 2, then 9 is an odd number.“

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Converse 2

Converse (2)

  • Is the Converse of a given condition logically equivalent to the Condition?

  • The truth table for the proposition & its converse are p  q & q  p

  • The two propositions are not logically equivalent

T

T

T

T

F

T

F

T

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Inverse 1

Inverse (1)

  • The inverse of a conditional statement is formed by negating the hypothesis and negating the conclusion of the original statement. In other words, the word "not" is added to both parts of the sentence.

  • Conditional:  "If 9 is an odd number, then 9 is divisible by 2.“

  • Inverse:  "If 9 is not an odd number, then 9 is not divisible by 2.“

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Inverse 2

Inverse (2)

  • Is the Inverse of a given condition logically equivalent to the Condition? Or is it logically equivalent to the Converse of the condition?

T

T

T

T

T

T

F

F

T

T

T

F

T

T

F

F

F

F

T

T

T

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Contrapositive

Contrapositive

  • The contrapositive of a conditional statement is formed by

    • negatingboth the hypothesis and the conclusion, and then

    • interchanging the resulting negations.

  • In other words, the contrapositive negates and switches the parts of the sentence. 

  • It does BOTH the jobs of the INVERSE and the CONVERSE

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Example contrapositive

Example: Contrapositive

  • Conditional:  "If 9 is an odd number, then 9 is divisible by 2.“

  • Contrapositive:  "If 9 is not divisible by 2, then 9 is notan odd number."

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Contrapositive1

Contrapositive

  • The Contrapositive of the proposition

    p  q is ~q  ~p are logically equivalent.

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Double implication

Double implication

  • The double implication “p if and only if q” is defined in symbols as p  q

    p  q is logically equivalent to (p q)^(q  p)

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Tautology

Tautology

  • A proposition is a tautology if its truth table contains only true values for every case

    • Example: p  p v q

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Contradiction

Contradiction

  • A proposition is a contradiction if its truth table contains only false values for every case

    • Example: p ^ ~p

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De morgan s laws for logic

De Morgan’s laws for logic

  • The following pairs of propositions are logically equivalent:

    • ~ (p  q) and (~p) ^ (~q)

    • ~ (p ^ q) and (~p)  (~q)

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Distributive laws of propositions

Distributive laws of propositions

  • p  (q ^ r) = (p  q) ^ (p r)

  • p ^ (q  r) = (p ^ q)  (p ^ r)

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Precedence of propositions

Precedence of propositions

~ is the highest

^

↔ lowest

  • Example: The proposition

    ~ p ^ q  r ↔ q  p ^ r

    is equivalent to or is computed according to:

    (((~p) ^ q)  r) ↔ ( q  (p ^ r))

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Proofs

A mathematical system consists of

Undefined terms

Definitions

Axioms

Proofs

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Undefined terms

Undefined terms are the basic building blocks of a mathematical system. These are words that are accepted as starting concepts of a mathematical system.

Example: in Euclidean geometry we have undefined terms such as

Point

Line

Undefined terms

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Definitions

Definitions

  • A definition is a proposition constructed from undefined terms and previously accepted concepts in order to create a new concept.

  • Example. In Euclidean geometry the following are definitions:

  • Two triangles are congruent if their vertices can be paired so that the corresponding sides are equal and so are the corresponding angles.

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Axioms

Axioms

  • An axiom is a proposition accepted as true without proof within the mathematical system.

  • There are many examples of axioms in mathematics:

    • Example: In Euclidean geometry the following are axioms

      • Given two distinct points, there is exactly one line that contains them.

      • Given a line and a point not on the line, there is exactly one line through the point which is parallel to the line.

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Theorems

Theorems

  • A theorem is a proposition of the form

    p  q

    which must be shown to be true by a sequence

    of logical steps that assume that p is true, and

    use definitions, axioms and previously proven

    theorems.

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Lemmas and corollaries

A lemma is a small theorem which is used to prove a more general theorem.

A corollary is a theorem that can be proven to be a logical consequence of a general theorem.

Example from Euclidean geometry: "If the three sides of a triangle have equal length, then its angles also have equal measure."

Lemmas and corollaries

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Types of proof

Types of proof

  • A proof is a logical argument that consists of a series of steps using propositions in such a way that the truth of the theorem is established.

  • Direct proof: p  q

    • A direct method of attack that assumes the truth of proposition p, axioms and proven theorems so that the truth of proposition q is obtained.

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Indirect proof

Indirect proof

There are two ways of indirect proofs:

  • The method of proof by contradiction of a theorem p  q consists of the following steps:

    1. Assume p is true and q is false

    2. Show that ~p is also true.

    3. Then we have that p ^ (~p) is true.

    4. But this is impossible, since the statement p ^ (~p) is always false. There is a contradiction!

    5. So, q cannot be false and therefore it is true.

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Indirect proof1

Indirect proof

  • The method of proof by showing that the contrapositive (~q)  (~p) is true.

    • Since (~q)  (~p) is logically equivalent to

      p  q,

      then the theorem is proved.

    • Example: show that  is a subset of every set.

      • From the definition of subsets, X Y if every element of X is also contained in Y, let p stands for element in X and q element in Y. Then to show

        p  q,

        we show (~q)  (~p). That is if the element is not in Y it is not in X. But there are no elements in the empty set,

        thus ~p is always true and hence (~q)  (~p) is true. Therefore p  q is true.

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Valid arguments

Valid arguments

  • Deductive reasoning: the process of reaching a conclusion q from a sequence of propositions p1, p2, …, pn.

  • The propositions p1, p2, …, pn are called premises or hypothesis.

  • The proposition q that is logically obtained through the process is called the conclusion.

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Rules of inference 1

1. Law of detachment or modus ponens

p  q

p

Therefore, q

2. Modus tollens

p  q

~q

Therefore, ~p

Rules of inference (1)

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Rules of inference 2

3. Rule of Addition

p

Therefore, p  q

4. Rule of simplification

p ^ q

Therefore, p

5. Rule of conjunction

p

q

Therefore, p ^ q

Rules of inference (2)

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Rules of inference 3

6. Rule of hypothetical syllogism

p  q

q  r

Therefore, p  r

Rules of inference (3)

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Rules of inference 4

Rules of inference (4)

7. Rule of disjunctive syllogism

  • p  q

  • ~p

  • Therefore, q

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Resolution proofs

Resolution proofs

  • A clause is a compound statement with terms separated by “or”, and each term is a single variable or the negation of a single variable

    • Example: p  q  (~r) is a clause

      (p ^ q)  r  (~s) is not a clause

  • In proving a statement, the hypothesis and conclusion are written as clauses

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Proof only one rule example

Proof Only one rule example

  • Example:

    • p  q

    • ~p  r

    • Therefore, q  r

      Its proof follows:

    • p  q Ξ~~p  q Ξ~p q Ξ ~q  p Using contrapositive

    • ~p  r Ξp  r

    • 1 & 2 ~q  p, rule 6, & p  r implies

      ~q  r

    • ~q  r Ξ q  r

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Example

Example

  • Let p, q, r & s be four propositions such that

    p v q, p  r, q  s therefore r v s

  • Assume ~ (r v s)= ~r ^ ~s Thus ~r & ~s are true from rule 4

  • ~p v r from p  r

  • ~q v s from q  s

  • From 1 ~r, ~s

  • From 4, 2 and rule 7 ~p

  • From 4, 3 and rule 7 ~q

  • Therefore ~(pvq)

  • Contradicting 5 & 6 p v q

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Example1

Example

  • Given p  (p q)

    p

    show q and p  q

  • p  (p q) ↔ ~ p  (p q) ↔ ~ p  (~ p  q) ↔ ~ p  ~ p  q ↔ ~ p  q

  • P

  • From 1, 2, and rule 7 q

  • From 2, 3 since p & q then p  q

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Predicate logic

Predicate Logic

  • A predicate is that part of a sentence which states something about the object of the sentence.

  • A predicate is a statement with a place for an object. When this place is filled, the predicate becomes a statement about the object that fills it.

  • A predicate is a proposition with a hole in it this hole is a variable.

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Predicates propositions

Predicates & Propositions

  • A statement such as x > 5 is not a proposition: its truth depends upon the value of variable x.

  • Before we can reason about such statements, we will need to declare, or introduce, the variables concerned.

  • The declaration x : a introduces a variable x and tells us that it is an element of the set a.

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For every and for some

For every and for some

  • Most statements in mathematics and computer science use terms such as for every and for some.

  • For example:

    • For every triangle T, the sum of the angles of T is 180 degrees.

    • For every integer n, n is less than p, for some prime number p.

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Universal quantifier

Universal quantifier

  • One can write P(x) for every x in a domain D

    • In symbols: x ε D, P(x)

  •  is called the universal quantifier

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Truth of as propositional function

Truth of  as propositional function

  • The statement x P(x) is

    • True if P(x) is true for every x  D

    • False if P(x) is not true for some x  D

  • Example: Let P(n) be the propositional function n2 + 2n is an odd integer

    n  D = {all integers}

  • P(n) is true only when n is an odd integer, false if n is an even integer.

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Existential quantifier

Existential quantifier

  • For some x  D, P(x) is true if there exists an element x in the domain D for which P(x) is true. In symbols: x, P(x)

  • The symbol  is called the existential quantifier.

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Examples

Examples

  • Let the universe set be the positive integers, and

    • let s(x) be the predicate “x is even integer”,

    • t(x,y) be “x = 2y”, p(x) be “x is a prime integer”, and

    • r(x) for “x >2”

      then we can rewrite the following statements as predicates:

  • “If x is an even integer, then there is an integer y such that x = 2y” can be written as

    • s(x)  y (t(x,y))

    • What is the truth value of the expression?

    • TRUE

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More examples

More Examples

  • “There is a prime integer x such that x = 2y for some y” can be written

    • x(p(x) ^ y (t(x,y)))

    • What is the truth value of the expression?

    • TRUE

  • “For all integers x, if x is a prime, then

    if x>2, then x is not an even integer” can be written

    • x (p(x)  (r(x)  ~s(x)))

    • What is the truth value of the expression?

    • TRUE

  • Discrete math


    Free bound variables

    Free & bound variables

    • In a predicate if a variable is preceded by a quantifier, then it is bounded by the quantifier otherwise it is free variable the portion of the expression for which the bound is applied is called the scope of the quantifier

      Examples:

      • In s(x) y (t(x,y)) y is bounded & x is free

        could be interpreted in only one way

      • Now consider x p(x) ^ y (t(x,y))

        In this expression x is bounded in x p(x) & y is free.

        But in y (t(x,y)) y is bounded and x is free.

      • Adding the parenthesis make the expression x(p(x) ^ y (t(x,y))) interpreted in a different way

      • In the expression x (p(x)  (r(x)  ~s(x))) the scope of x is the (p(x)  (r(x)  ~s(x))) i.e. we talk about the same x

    Discrete math


    Understanding predicate expressions

    Understanding predicate expressions

    • Rewrite the following predicate expressions in English:

      • x (p(x) ^ y (t(x,y) ^ r(x)))

        For each integer x, x is prime, and there is an integer y such that x is 2y and x >2

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    Counterexample

    Counterexample

    • The universal statement x P(x) is false if x  D such that P(x) is false.

    • The value x that makes P(x) false is called a counterexample to the statement x P(x).

      • Example: P(x) = "every x is a prime number", for every integer x.

      • But if x = 4 (an integer) this x is not a prime number. Then 4 is a counterexample to P(x) being true.

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    Rules of inference for quantified statements

    1. Universal instantiation

     xD, P(x)

    d  D

    Therefore P(d)

    2. Universal generalization

    P(d) for any d  D

    Therefore x, P(x)

    3. Existential instantiation

     x  D, P(x)

    Therefore P(d) for some d D

    4. Existential generalization

    P(d) for some d D

    Therefore  x, P(x)

    Rules of inference for quantified statements

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    Generalized de morgan s laws for logic

    Generalized De Morgan’s laws for Logic

    • If P(x) is a propositional function, then each pair of propositions in a) and b) below have the same truth values:

      a) ~(x P(x)) and x ~P(x)

      "It is not true that for every x, P(x) holds" is equivalent to "There exists an x for which P(x) is not true“

      b) ~(x P(x)) and x ~P(x)

      "It is not true that there exists an x for which P(x) is true" is equivalent to "For all x, P(x) is not true"

    Discrete math


    Nested quantifiers 1

    Nested quantifiers(1)

    • In each of the most cases, an individual quantifier applies to a single variable.

    • Now consider the example:

      • Assume that the universe of discourse is the set of all nonzero real numbers. Then,

      • the statement “for all x, there exists a y such that x + y = 1" can be written as

      • xyP (x; y) where P (x; y) is the predicate x + y = 1.

      • Note that separate quantifiers are used for x and y.

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    Nested quantifiers 11

    Nested quantifiers(1)

    • We refer to a sequence of quantifiers applied to a predicate P as nested quantifiers because each quantifier applies to a predicate obtained by applying other quantifiers in the sequence to the original predicate P.

    • For example, in the proposition xy P (x; y), the outermost quantifier, x,is actually applied to the predicate y P (x; y).

    • This implies that quantifiers themselves can be affected by other quantifiers; that is, nested quantifiers applied to the same predicate are not applied independently of one another, even if they are applied to different variables.

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    The order of quantifiers

    The Order of Quantifiers

    • The proposition xyP (x; y) is equivalent to the proposition yxP (x; y).

      • That is, the order in which universal quantifiers are applied to different variables is irrelevant.

    • Similarly, the order in which existential quantifiers are applied to different variables does not affect the resulting proposition.

    • However, the order in which quantifiers are applied is important when existential and universal quantifiers are mixed, as the following example shows.

    Discrete math


    Example on order of quantifiers

    Example on order of quantifiers

    • Consider the proposition x y P(x; y).

      • This proposition is true if, for every x, y P(x; y) is true.

      • That is there exists at least one value of y for which P(x; y) is true.

      • The proposition is false if there exists any value of x for which yP (x; y) is false. But what make yP (x; y) false?

      • if there exists any value of x for which P(x; y) is false for every y.

    Discrete math


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