Chapter 1: Defining Sets

Question 0

X∈Y

Answer 0

X is a member of Y

Question 1

What is a set?

Answer 1

A set is a collection of arbitrary objects, where no object has multiple occurrences, and the order of the objects doesn’t matter.

Question 2

∀x∈X

Answer 2

For every x in X.

Question 3

∃x∈X

Answer 3

There exists x in X.

Question 4

Logical Axioms

Answer 4

Axioms of logic, most commonly, first-order classical logic with equality. Logical axioms are shared by all theories.

Question 5

Specific Axioms

Answer 5

Axioms specific to a given theory, such as the Axiom of Pairing.

Question 6

Axiom of Pairing

Answer 6

For all objects x1,x2, there exists a set Y containing exactly x1,x2 as members.

Question 7

Axiom of Extensionality

Answer 7

Sets X,Y are equal, denoted X=Y, if and only if they have the same elements.

Question 8

The singleton of x

Answer 8

The set that has x as its only member. Denoted {x}.

Question 9

Finite Set

Answer 9

A set that has n elements, for some natural number n.

Question 10

Empty or Null set

Answer 10

A set with no members. Denoted { }.

Question 11

X ⊆ Y

Answer 11

X is a subset of Y, meaning every element of X is also an element of Y.

Question 12

X ⊂ Y

Answer 12

X is a proper subset of Y, meaning X consists of some elements of Y, but not all.

Question 13

Z

Answer 13

Stands for the set of all integers

Question 14

Z+

Answer 14

Stands for the set of all positive integers.

Question 15

N

Answer 15

Stands for the set of all natural numbers

Question 16

Q

Answer 16

Stands for the set of all rational numbers.

Question 17

R

Answer 17

Stands for the set of all real numbers.

Question 18

C

Answer 18

Stands for the set of all complex numbers.

Question 19

X ∪ Y

Answer 19

The union of X and Y, the set that contains all the elements which are in X or in Y, but no other elements.

Question 20

X ∩ Y

Answer 20

The intersection of X and Y. The set of the common parts of X and Y.