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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

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T

Discrete math

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.“

Discrete math

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

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Discrete math

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.“

Discrete math

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?

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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

Discrete math

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."

Discrete math

Contrapositive

- The Contrapositive of the proposition
p q is ~q ~p are logically equivalent.

Discrete math

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)

Discrete math

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

- A proposition is a contradiction if its truth table contains only false values for every case
- Example: p ^ ~p

Discrete math

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

- p (q ^ r) = (p q) ^ (p r)
- p ^ (q r) = (p ^ q) (p ^ r)

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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))

Discrete math

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 termsDiscrete math

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.

Discrete math

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.

- Example: In Euclidean geometry the following are axioms

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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.

Discrete math

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 corollariesDiscrete math

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.

Discrete math

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.

Discrete math

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.

- 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

- Since (~q) (~p) is logically equivalent to

Discrete math

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.

Discrete math

1. Law of detachment or modus ponens

p q

p

Therefore, q

2. Modus tollens

p q

~q

Therefore, ~p

Rules of inference (1)Discrete math

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)Discrete math

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

- Example: p q (~r) is a clause
- In proving a statement, the hypothesis and conclusion are written as clauses

Discrete math

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

Discrete math

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

Discrete math

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

Discrete math

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.

Discrete math

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.

Discrete math

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

- One can write P(x) for every x in a domain D
- In symbols: x ε D, P(x)

- is called the universal quantifier

Discrete math

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.

Discrete math

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.

Discrete math

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

Discrete math

More Examples “For all integers x, if x is a prime, then

- “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

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

- 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

- In s(x) y (t(x,y)) y is bounded & x is free

Discrete math

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

- x (p(x) ^ y (t(x,y) ^ r(x)))

Discrete math

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.

Discrete math

xD, 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 statementsDiscrete math

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)

- 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
- xyP (x; y) where P (x; y) is the predicate x + y = 1.
- Note that separate quantifiers are used for x and y.

Discrete math

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 xy 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.

Discrete math

The Order of Quantifiers

- The proposition xyP (x; y) is equivalent to the proposition yxP (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

- 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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