Lesson 2: Equivalences, Quantifiers

1
Lesson 2 Equivalences, Quantifiers Predicates

• Objectives
• Define the terms introduced in this lesson
• Prove equivalences using truth tables
• Prove equivalences using a step-by-step proof
• State the truth value of a propositional function
  given its arguments
• Determine truth values of universal
  quantifications and existential quantifications
• Explain the difference between a propositional
  function and a proposition

• Explain why the equivalences in Table 2 are true
• Translate logical expressions to and from English
• Outline
• Proving Equivalence
• Propositional Functions
• Quantifiers
• Reading Section 1.2, 1.3

2
Logical Equivalence

• If p?q is always true, then p and q are
  logically equivalent
• p?q

3
Definitions

• Consider a compound proposition.
• It is a
• Tautology if it is always true
• Contradiction if it is always false
• Contingency if it is neither (depends on values)

4
Proving Logical Equivalence

• Method 1 Truth Tables
• p?(q?r) ? (p ? q) ?(p ? r)
• Method 2 Step-by-Step
• Tables 5-7, page 24

5
In-class Exercise

• Prove p?q ? ?q ? ?p
• Using truth tables
• Using identities

6
Propositional Functions

• x 4
• y gt x
• x 8 10
• These statements are neither true nor false.
• We can denote such statements P(x) or P(x,y)
  since their truth value depends on the value of
  the variable
• Such statements are said to be propositional
  functions
• When the variable of a propositional function is
  assigned a value, then it becomes a proposition
  (and thus has a truth value T or F).

7
Propositional Functions

• What are the truth values of the following
  propositional functions
• Q(x,y) denotes the statement x is equal to y
• Q(2,3)
• Q(2,3)
• R(x,y,z) denotes the statement x2y2-z20
• R(5,12,13)
• R(2, 2, 4)

8
Universal Quantifier

• The given range of values a variable can take on
  is called the universe of discourse
• The universal quantification of P(x) is the
  proposition
• P(x) is true for every value of x in the universe
  of discourse

9
Universal Quantifier

• P(x) x 1 gt x (all real numbers)
• P(x) x x2
• (integers)
• T(x) can be the American president
• (people)
• (people born in the US)

10
Existential Qualifier

• There exists an element in the universe of
  discourse such that P(x) is true
• P(x) x x2
• (integers)
• T(x) Has a birthday in June
• (ece357 students)

11
Negations

• Negation of the existential quantifier is the
  universal quantifier of the negation of the
  propositional function
• Negation of the universal quantifier is the
  existential quantifier of the negation of the
  propositional function