Sets and Their Properties

Study Notes

Power Sets

A power set (2^S) is the set of all subsets of a given set (S).
Example: The power set of {a, b} is {{}, {a}, {b}, {a, b}}.
The empty set's power set is {{}}.
The size of the power set (|2^S|) is 2^|S|, where |S| is the size of set S.

Partitions

A partition of a set (S) is a collection of non-overlapping subsets that together cover the entire set.
Each element in S must be in exactly one subset of the partition.
The empty set (∅) is not part of a partition.
S can be partitioned into subsets. e.g. {{a,c}, {b,d,e}, {f}} is a partition of {a,b,c,d,e,f}.
A set can be partitioned into itself {{a,b,c,d,e,f}} is a partition of {a,b,c,d,e,f}.
A set can be partitioned into sub-sets {{a,b,c}, {d,e,f}} is a partition of {a,b,c,d,e,f}.

Ordered Pairs

An ordered pair (x, y) is a pair where the order of the elements matters.
(x, y) ≠ (y, x) if x ≠ y.

Cartesian Product

The Cartesian product (A x B) of two sets A and B is the set of all ordered pairs (x, y) where x is in A and y is in B.
Example: {a} x {b, c} = {(a, b), (a, c)}.
The Cartesian product can also be extended to more than two sets—for example, A x B x C results in ordered triples.
Example: {a} x {b, c} x {d} = {(a, b, d), (a, c, d)}.
The Cartesian product is often presented using nested parentheses (e.g., (A x B) x C); however, often the outer parentheses can be omitted for clarity and readabilty.

Description

Explore the concepts of power sets, partitions, ordered pairs, and Cartesian products in set theory. This quiz will test your understanding of how subsets are formed, the uniqueness of ordered pairs, and the nature of set combinations. Perfect for students studying mathematics and set theory.