07 Negation of Quantified Statements + Statements with Multiple Quantifiers

some (Alternative answers: at least one; an); does not have property S

A negation for "All R have property S" is "There is _____ R that _____ ."

There is an x such that x has property P and x does not have property Q

A negation for "For all x, if x has property P then x has property Q" is " _______."

No R have property S

A negation for "Some R have property S" is " __________."

The negation of a statement of the form ∀x in D, Q(x) is logically equivalent to a statement of the form ∃x in D such that ∼Q(x). Symbolically, ∼(∀x ∈ D, Q(x)) ≡ ∃x ∈ D such that ∼Q(x).

Negation of a Universal Statement

The negation of a statement of the form ∃x in D such that Q(x) is logically equivalent to a statement of the form ∀x in D, ∼Q(x). Symbolically, ∼(∃x ∈ D such that Q(x)) ≡ ∀x ∈ D, ∼Q(x).

Negation of an Existential Statement

Negations of Multiply-Quantified Statements ∼(∀ x in D, ∃y in E such that P(x, y)) ≡ ∃x in D such that ∀y in E, ∼P(x, y). ∼(∃x in D such that ∀y in E, P(x, y)) ≡ ∀x in D, ∃y in E such that ∼P(x, y).

Negations of Multiply-Quantified Statements

∼(∀x, if P(x) then Q(x)) ≡ ∃x such that P(x) ^ ∼Q(x).

Negations of Universal Conditional Statements

True. Imagine any positive real number x on the real number line. These numbers correspond to all the points to the right of 0. Observe that no matter how small x is, the number x/2 will be both positive and less than x.∗

There Is No Smallest Positive Real Number True of False?

True. 1 is a positive integer that is less than or equal to every positive integer.

There Is a Smallest Positive Integer: ∃ a positive integer m such that ∀ positive integers n, m ≤ n. True of False?

an element y in E; y; P(x, y)

To establish the truth of a statement of the form "∀x in D, ∃y in E such that P(x, y)," you imagine that someone has given you an element x from D but that you have no control over what that element is. Then you need to find _______ with the property that the x the person gave you together with the ________ you subsequently found satisfy _______

an element x in D; y in E; P(x, y)

To establish the truth of a statement of the form "∃x in D such that ∀y in E, P(x, y)," you need to find ___ so that no matter what ____ a person might subsequently give you, _________ will be true.