Set theory

Question 1

Which axioms does naive set theory entail?

Answer 1

The axiom of extensionality and the axiom of comprehension.

Question 2

Explain the axiom of extensionality.

Answer 2

For sets A and B: A=B if and only if (for any x) (x belongs to A iff x belongs to B)

Question 3

Explain the axiom of comprehension

Answer 3

For any condition C, there exists a set A such that (for any x) (x belongs to A iff x satisfies C).

Question 4

What is Russells paradox?

Answer 4

The paradox that arises from the axiom of comprehension: through proof by contradiction one can show both that the set R (Russell set) is a a member of itself an is not a member of itself.

Question 5

What is Russells set?

Answer 5

The set with set condition ‘is not a member of itself’.

Question 6

What is a power set? How many members does a power set have?

Answer 6

A power set is a set of all the subsets of a set. A power set of a set with n members has 2^n subsets.

Question 7

Power set theorem

Answer 7

The power set of a set has a strictly greater cardinality than itself.

Question 8

infinite set

Answer 8

a set is infinite if it can be put in one-to-one correspondence with a proper subset of it.