Express the negation of each of these statements in terms of quantifiers without using the negat

Express the negation of each of these statements in terms of...

Key Concepts

Negation of Quantified Statements

This concept involves the logical equivalences that allow one to switch between universal and existential quantifiers when negating predicates. In particular, the negation of a statement of the form 'for all x, P(x)' is logically equivalent to 'there exists an x such that not P(x)', and the negation of 'there exists an x such that P(x)' is equivalent to 'for all x, not P(x)'. Using these equivalences, we can express the negation of quantified statements without having an explicit negation symbol on the outside.

De Morgan's Laws for Logical Expressions

De Morgan's Laws are rules in logic that provide relationships between conjunctions and disjunctions under negation. They state that the negation of a disjunction is equivalent to the conjunction of the negations, and the negation of a conjunction is equivalent to the disjunction of the negations. These rules are essential when translating the negation of compound predicates, particularly when the predicate includes logical or expressions, as they allow for the appropriate rewriting in terms of quantifiers and connectives.

Negation of Relations

When negating numerical or relational statements, such as inequalities, the direction and type of the inequality often change. For example, the negation of an inequality like 'x > a' becomes 'x ? a', and similarly, 'x ? a' negates to 'x < a'. This concept is crucial when removing the negation operator from quantified statements that involve relational expressions, ensuring that the meaning of the predicate is correctly inverted.

Transcript

00:01 So let's see, we have a couple of statements here that we'd like to negate without actually using the negation symbol not.

00:10 To do that i'm going to first make this a little bit skinnier, and then we'll note here that the negation of for all x something is true of x is precisely there exists some x such that phi of x is not a lie.

00:29 So if it is not true that every x has some property that is the same thing as saying that there is some x that does not have the property and conversely if there does not exist an x some property that is the same thing as saying that all x do not have that property.

00:51 So we use those two things and the negations of these formulas to do what we want here.

00:55 A, so we have for all x, excuse me, for all x x is greater than one, for all x x is greater than one.

01:05 To negate that we'll do there exists some x where x is not greater than one, and in particular x is not greater than one means that x is less than or equal to one.

01:19 And there we go.

01:21 For part b our statement is for all x, i keep doing my parentheses too soon, for all x x is less than or equal to two.

01:30 We negate that means there exists an x such that x is not less than or equal to two, and if x is not less than or equal to two x must be greater than two.

01:46 C here, there exists an x that is greater than or equal to four.

01:53 To negate that we say for all x x is not greater than or equal to four.

01:58 What does it mean for x not to be greater than or equal to four? well it means that x has to be less than four.

02:05 D here, there exists an x such that x is less than zero.

02:11 We negate that means for all x x is not less than zero.

02:16 What is not less than zero? why? things that are greater than or equal to zero of course.

02:22 Part e, we're almost done...

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