Logic of Quantified Statements: Multiple Quantifiers and Negations, Slides of Discrete Mathematics

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Chapter 2

The Logic of Quantified Statements

Section 2.

Multiple Quantifiers

∀∃ Statement

• Show that the following

is true: “For all triangles in x, there is a square y such that x and y have the same color.”

∃∀ Statement

• Show that the following

is true: “There is a triangle x such that for all circles y, x is to the right of y.”

Informal to Formal Examples

• A reciprocal of a real number a is a real number b

such that ab = 1.

• Translate the following

• Every nonzero real number has a reciprocal.
• ∀nonzero real number u, ∃a real number v such that

u*v = 1.

• There is a real number with no reciprocal.
• ∃a real number c such that ∀real numbers d, c*d ≠ 1

Example

• Translate the following to formal:

• “There is a smallest positive number.”
• ∃a positive integer m such that ∀ positive integers n, m ≤ n.
• Is this true? Is there a positive integer such that it is equal or less than any other positive integer.
• Yes…. 1 works.

Example

• Write a negation for the following

statements:

• For all squares x, there is a circle y such that x and y have the same color. - First: ∃a square x such that ~(∃a circle y such that x and y have the same color) - Final: ∃a square x such that ∀circles y, x and y do not have the same color. - Negation is true. sq e is black and no circle is black.

Example

• Write a negation for the following

statements:

• There is a triangle x such that for all squares y, x is to the right of y. - First: ∀triangles x, ~(∀squares y, x is to the right of y) - Final: ∀triangles x, ∃ a square y such that x is not to the right of y.