Quantifiers and Cosine Function Concepts

The video introduces predicates and quantifiers in mathematics, explaining how variables are used in mathematical sentences. It covers existential and universal quantifiers, their symbols, and how to determine the truth of quantified statements. The video also discusses negating quantified statements and provides examples to illustrate these concepts.

What is a predicate in mathematical sentences?

• A sentence that is always true

• A sentence that contains a variable

• A sentence without a variable

• A sentence with a specified variable

Which symbol represents the existential quantifier?

• An upside-down capital E

• A capital E facing the right direction

• A capital E facing the wrong direction

• An upside-down capital A

What does the universal quantifier assert?

• There exists at least one

• For all or every

• Some exist

• None exist

In discrete mathematics, what is the usual domain of discourse?

• All elements of any set

• Natural numbers

• All positive integers

• All real numbers

What happens to quantifiers when a statement is negated?

• They become universal

• They switch type

• They disappear

• They remain the same

What is the negation of 'for every x, there exists a y such that y is less than x'?

• There exists an x for every y such that y is greater than or equal to x

• For every y, there exists an x such that y is greater than x

• There exists a y for every x such that y is less than x

• For every x, there exists a y such that y is greater than or equal to x

In the context of cosine function, what is true for every x?

• Cosine x is always greater than 1

• There exists a y such that cosine x equals y

• Cosine x equals y for no y

• Cosine x is always less than -1

Why is the statement 'for every y, there exists an x such that cosine x equals y' false?

• Because y is always on the closed interval from -1 to 1

• Because cosine x can be any real number

• Because y is always a real number

• Because x is not a real number

What is the range of the cosine function?

• From -1 to 0

• From -1 to 1

• From 0 to 1

• From -2 to 2

What does the statement 'there exists a y for every x such that cosine x doesn't equal y' imply?

• Cosine x equals y for all y

• Cosine x doesn't equal y for some y

• Cosine x equals y for no y

• Cosine x doesn't equal y for all y