80-110 The Nature of Mathematical Reasoning

Thursday, April 18, 2001

Quiz 10

Name: _______________________________

1. Who is the founder of modern set theory?

2. Georg Cantor (1845-1918)

3. What does ZF stand for?

4. The modern axioms of set theory, due to Zermelo and Fraenkel.

5. What does it mean for a function f:A->B to be 1-1 (one-to-one, injective)?

6. No two different elements of the set A are mapped to the same element in B.

7. Which of the following two sentences of predicate logic can be true in a finite domain, and which in an infinite domain of objects: "forall x exists y ( x < y )" and "exists x forall y ( x < y )"?

8. "forall x exists y ( x < y )" is true if for all numbers there exists a greater number. This can be true only with infinite sets and it certainly the case for the natural numbers.

"exists x forall y ( x < y )" is true if there is one number such that all numbers are greater than it. This sentence is false, since no number is greater than itself. If would be true if the relation were "greater or equal", because in this case there is a number such that all numbers are greater or equal, namely 1.