Chapter 1: Logic and Proofs

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Chapter 1 Logic and Proofs

• Discrete Mathematics
• ???

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What is the Logic?

• Logic is the study of reasoning it is
  specifically concerned with whether reasoning is
  correct.
• Logic focuses on the relationship among
  statements as opposed to the content of any
  particular statement. For example
• All mathematicians wear sandals.
• Anyone who wears sandals are algebraists.
• Therefore, all mathematicians are algebraists.

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Propositions

• A proposition is typically expressed as a
  declarative sentence (as opposed to a question,
  command, etc.).
• Propositions are the basic building blocks of any
  theory and logic.
• A proposition can be expressed as
• p 11 3

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Definition Binary Operators

• Let p and q be propositions
• The conjunction of p and q, denoted
• is the proposition p and q.
• The disjunction of p and q, denoted
• is the proposition p or q.

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Example 1.1.2

• If
• p It is raining.
• q It is cold.
• Then
• The conjunction of p and q is
• It is raining and cold.
• The disjunction of p and q is
• It is raining or cold.

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Definition Truth Value

• The truth value of the proposition and
• are defined by the truth table as follows

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Definition Negation Operator

• The negation of p, denoted , is the
  proposition not p and the truth table of
• is as follows

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Example 1.1.10

• If
• p pwas calculated to 1,000,000 decimal digits
  in 1954.
• Then, the negation of p is the proposition
• pwas not calculated to 1,000,000 decimal
  digits in 1954.

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Conditional Propositions

• If p and q are propositions, the proposition if
  p then q is called a conditional proposition and
  is denoted .
• The proposition p is called the hypothesis (or
  antecedent) and the proposition q is called the
  conclusion (or consequent).
• Another expression of conditional proposition is

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Truth Value of the Conditional Proposition

• The truth value of the conditional proposition
• is defined by the following

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Operator Precedence

• In expressions involving some or all of the
  operators , , and , in the
  absence of parentheses, we first evaluate ,
  then , then , and then .
• Given that proposition p is false, proposition q
  is true, and proposition r is false, what is the
  truth value of the proposition
  ?(Ex. 1.1.12)
• What is the truth value of
  ?

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Sufficient and Necessary Conditions

• A sufficient condition is a condition that
  suffices to guarantee a particular outcome.
• If the condition does not hold, the outcome might
  be achieved in other ways or it might not be
  achieved at all but if the condition does hold,
  the outcome is guaranteed.
• Similarly, a necessary condition is a condition
  that is necessary for a particular outcome to be
  achieved.
• The condition does not guarantee the outcome
  but, if it does not hold, the outcome will not be
  achieved.

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Example 1.2.6

• (d) A necessary condition for the Cubs to win the
  World Series is that they sign a right-handed
  relief pitcher.
• (e) A sufficient condition for Maria to visit
  France is that she goes to the Eiffel Tower.

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Biconditional Proposition

• If p and q are propositions, the proposition
• p if only if q is called a biconditional
  proposition and is denoted .
• An alternative way to state p if only if q is
  p is a necessary and sufficient condition for
  q, and sometimes can be writtens as p iff q.
• Equivalently,
  .

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The Truth Value of Biconditional Propositions
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Logically Equivalent

• Suppose that the propositions P and Q are made up
  of the propositions . We
  say that P and Q are logically equivalent and
  write , provides that, given any
  truth values of , either P
  and Q are both true, or P and Q are both false.

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De Morgans Law for Logic
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Example 1.2.13

• Show that the negation of is
• Logically equivalent to

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Theorem 1.2.18

• The conditional proposition and
• its contrapositive are
  logically
• equivalent.
• Proof omitted.

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The Domain of Discourse

• Let P(x) be a statement involving the variable x
  and let D be a set. We call P a propositional
  function or predicate (with respect to D) if for
  each x in D, P(x) is a proposition. We call D the
  domain of discourse of P.
• A propositional function P, by itself, is neither
  true nor false. However, for each x in its domain
  of discourse, P(x) is a proposition and is either
  true or false.
• For example, let P(n) be the statement
• P(x) n is an odd integer
• and let D be the set of positive integers. Then
  P is a propositional function with domain of
  discourse D since for each n in D, P(n) is a
  proposition.

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Universally Quantified Statement and Universal
Quantifier

• Let P be a propositional function with domain of
  discourse D. The statement for every x, P(x)
  (also denoted P(x)) is said to be
  universally quantified statement. The symbol
• is called a universal quantifier.
• P(x) is true if P(x) is true for every x
  in D.
• P(x) is false if P(x) is false for at
  least one x in D.

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Existentially Quantified Statement and
Existential Quantifier

• Let P be a propositional function with domain of
  discourse D. The statement there exists x, P(x)
  (also denoted P(x)) is said to be
  existentially quantified statement. The symbol
• is called a existential quantifier.
• P(x) is true if P(x) is true for at least
  one x in D.
• P(x) is false if P(x) is false for every
  x in D.

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Example 1.3.8 and 1.3.11

• To prove the universally quantified statement
  for every real number x, if x gt 1, then x 1 gt
  1 is true.
• To verify that the existentially quantified
• statement is
  false.

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Generalized De Morgan Laws for Logic

• If P is a propositional function, each pair of
• propositions in (a) and (b) has the same
• truth values.
• (a)
• (b)

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Nested Quantifiers

• Consider the statement the sum of any two
  positive real number is positive can be
  expressed as for every x and for every y, if x
  gt 0 and y gt 0, then x y gt 0, also can
  symbolically expressed as
• Multiple quantifiers such as are said
  to be nested quantifiers.

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Example 1.4.2

• Write the assertion Everybody loves somebody
  symbolically.
• Hint letting L(x,y) x loves y

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Example 1.4.11 and 1.4.12

• Consider the statements
• (a)
• (b)
• with domain of discourse the set of positive
  integers.

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Example 1.4.13

• Using the generalized De Morgan Laws for
• Logic to find what is the negation of

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Proofs

• Techniques of Proof
• Direct Proof Ex. 1.5.10
• Proof by Contradiction (Indirect Proof)
• Ex. 1.5.12
• Proof by Contrapositive Ex. 1.5.13
• Proof by Cases Ex. 1.5.14

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Deductive Reasoning

• The deductive reasoning is the process that draws
  a conclusions from a sequence of propositions.
• The given propositions are called hypotheses or
  premises.
• The proposition follows from the hypotheses is
  called conclusion.

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Argument

• A (deductive) argument consists of hypotheses
  together with a conclusion.
• Any argument has the form
• If p1 and p2 and .and pn, then q.
• Symbolicaly denoted as
• Rules of Inference for Proposition (pp. 45)
• Rules of Inference for Quantified statements
  (pp.46)

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Principle of Mathematical Induction

• Suppose that we have a propositional
• function S(n) whose domain of discourse is
• the set of positive integers. Suppose that
• S(1) is true
• ,if S(n) is true, then S(n1) is true.
• Then S(n) is true for every positive n.

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Example 1.7.3

• Prove that
• for all ,

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Strong Form of Mathematical Induction

• Suppose that we have a propositional
• function S(n) whose domain of discourse is
• the set of integers greater than or equal
• to n0. Suppose that
• S(n0) is true
• for all ngtn0, if S(k) is true for all k,
  ,
• then S(n) is true.
• Then S(n) is true for every integer .