15 Questions
Which quantifier is used to denote 'for all' in mathematical statements?
∀
In which type of proofs are the results established in contrapositive form?
Indirect proofs
Which connectives are commonly used in mathematical statements involving quantifiers?
∃ and ∨
Which of the following statements uses the existential quantifier?
There exists a prime number greater than 100.
What percentage of proofs in mathematics are typically either proofs by contradiction or in contrapositive form?
A very large percentage
What is the observation about the dependency of a quantifier mentioned in the text?
The quantifier depends on a set to make sense.
Which of the following best represents the negation of the statement 'There exists an element in a set X which has some property P'?
For each element of the set, we prove the element does not have the property P
What is the negation of the statement 'Every page in this book contains at least 500 words'?
Some pages in this book contain less than 500 words
In terms of quantifiers, what is the negation of '∃r ∈ Q (r2 = 2)'?
∀r ∈ Q (r2 ≠ 2)
What does it mean to say that the statement 'A nonempty subset A ⊂ R is bounded above in R' is false?
A is not bounded above in R
What is the negation of the statement 'Every student in the classroom is at least 5 feet tall'?
Every student in the classroom is less than 5 feet tall
In terms of quantifiers, what is the negation of '∃x ∈ X (x has property P )'?
∃x ∈ X (x does not have property P )
What would be the negation of the statement 'There exists a real number x such that x2 = 1'?
No real number x satisfies x2 = 1
What is the negation of the statement 'Every apple in the basket is ripe'?
Some apples in the basket are not ripe
If the statement 'There exists an element in a set X which has some property P' is false, what pattern does its negation follow?
'For each element of the set, we prove the element does not have the property P'
This quiz covers the basics of commonsense logic, the use of quantifiers in mathematical statements, and the different kinds of proof. It focuses on statements involving quantifiers ∀, ∃, connectives and/or, and the negation of sentences involving these.
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