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.

    1. 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.

    1. 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.

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