Week 1: Sets and Numbers

Question 1

Extensive Notation

Answer 1

Specifying a set by listing its members one by one, e.g., {James, Daniel, Robert}.

Question 2

Intensive Notation

Answer 2

Specifying a set by stating a condition fulfilled by all and only its members, e.g., {x: x was a cast member of The Walking Dead}.

Question 3

Membership notation

Answer 3

We use the symbol ∈ to denote “is a member of”.

Question 4

Axiom of Extensionality

Answer 4

Sets A and B are one and the same set if and only if every member of A is a member of B, and vice versa.

Question 5

Axiom of Comprehension

Answer 5

For any condition C, there is a set A of things that fulfill condition C.

Question 6

Union of sets

Answer 6

The union of sets A and B, noted A ∪ B, is the set of things that are in A or in B.

Question 7

Intersection of sets

Answer 7

The intersection of sets A and B, noted A ∩ B, is the set of things that are in both A and B.

Question 8

Empty set

Answer 8

The empty set, denoted by ∅, is a set with no members.

Question 9

Subset

Answer 9

A set A is a subset of set B, denoted A ⊆ B, if and only if every member of A is also a member of B.

Question 10

Proper subset

Answer 10

A set A is a proper subset of set B, denoted A ⊂ B, if and only if every member of A is also a member of B, and A is not equal to B.

Question 11

Russell’s Paradox

Answer 11

A paradox that arises from considering the set of all sets that do not contain themselves as a member. It shows that naive set theory is incoherent.

Question 12

Pairing, empty set and union Axioms

Answer 12

Axioms that define the empty set, pairing of sets, and union of sets in set theory. They are basic principles used to construct and manipulate sets.

Question 13

Power set axiom

Answer 13

The power set of a set A, noted P(A), is the set of all subsets of A. The power set axiom states that for any set A, there exists a set P(A) which is the power set of A.

Question 14

Power set axiom

Answer 14

The power set of a set A, noted P(A), is the set of all subsets of A. The power set axiom states that for any set A, there exists a set P(A) which is the power set of A.

Question 15

Inclusion

Answer 15

A set A is included in a set B if and only if anything that is a member of A is a member of B. Inclusion refers to the relationship between sets, whereas membership refers to the relationship between an element and a set.