1-5_Quantified_Statements

1.5 Quantified Statements

Open Sentences

• Definition: A sentence that contains a variable is called an open sentence. It becomes a statement only when its variables are assigned specific values.

• Example: The statement “x = 5” is true when x is 7 and false when x is 2.

• Domain: The set of possible values that the variables in the open sentence can take.

• Truth Set: The set of all values from its domain that make the open sentence true when substituted into the variable.

• Notation: An open sentence with variable x is denoted by P(x). For multiple variables x1,...,xn, it is denoted by P(x1,...,xn).

• Example 1.5.2: Consider P(x) = (x - 3)^2 ≤ 1.

  • When the domain is Z (integers), the truth set is {2, 3, 4}.

  • When the domain is R (real numbers), the truth set is [2, 4].

Existence Statements

• Existential Quantifier: Phrases such as “there exists,” “there is,” and “for some” are called the existential quantifier and denoted by ∃.

• Meaning: An existence statement of the form ∃x ∈ S such that P(x) means "There exists x in S such that P(x)".

• Truth of Existence Statement: It is true if and only if the truth set of P(x) is not empty. It can be proven either by finding a specific value a in S such that P(a) is true, or by showing that at least one such value exists.

• Example 1.5.5:

  1. Determine the truth value of the following statements:

  • ∃x ∈ R such that x² = 1: True (x = 1, -1)

  • ∃x ∈ C such that x² = 1: True (x = i, -i)

  • ∃x ∈ [0, 1] such that x⁷ - 12x³ + 16x = 0: True by the Intermediate Value Theorem.

• Exercise 1.5.6: Determine the truth value of the existence statement: "There exist an even integer s and an odd integer t such that |s - 1| + |t - 2| ≤ 2."

Unique Existence Statements

• Unique Existential Quantifier: Phrases like “there exists a unique” and “there is exactly one” are called the unique existential quantifier and denoted by ∃!.

• Interpretation: The statement ∃!x ∈ S such that P(x) is true if and only if the truth set contains exactly one element.

• Example 1.5.8:

  1. ∃!x ∈ N such that x is even and prime: True (x = 2).

  2. ∃!x ∈ Z such that x² = 4: False (x = 2, -2).

Universal Statements

• Universal Quantifier: Phrases like “for every,” “for each,” and “for all” are called the universal quantifier and denoted by ∀.

• Meaning: A universal statement ∀x ∈ S, P(x) is true iff P(x) holds true for every element in S.

• Example 1.5.10:

  • ∀x ∈ R, x² > 0: False (0 is not in the truth set).

  • Examine P(x,y): xy + 2 is prime for ∀x ∈ S, ∀y ∈ T where S = {1, 3, 5} and T = {3, 9}.

Negation of Quantified Statements

• Theorem 1.5.19:

  1. The negation of a universal statement becomes an existence statement with the negated open sentence: ¬∀x ∈ S, P(x) ≡ ∃x ∈ S such that ¬P(x).

  2. The negation of an existence statement becomes a universal statement with the negated open sentence: ¬∃x ∈ S such that P(x) ≡ ∀x ∈ S, ¬P(x).

• Examples:

  1. Negation of "For every set A, A ∩ A = ∅": There exists a set A such that A ⊆ A.

Translating Sentences into Symbolic Sentences

• Example 1.5.15:

  1. For every odd positive prime x less than 10, x² + 4 is prime.

  • Symbolic: ∀x ∈ Z, (x is odd ∧ x is prime ∧ 0 < x < 10) → x² + 4 is prime.

  2. Some functions defined at 0 are not continuous at 0.

  • Symbolic: ∃f ∈ F such that f is defined at 0 ∧ f is not continuous at 0.

  3. Some integers are even and some are odd.

  • Symbolic: ∃x ∈ Z such that x is even ∧ ∃y ∈ Z such that y is odd.

Exercise Practice

• Negate Statements of Quantifiers:

  • Translate and determine the truth value of quantified statements such as "Every differentiable function is continuous" and others as per Exercises 1.5.16, 1.5.22.