- Sets, Sequences, and Ordered Tuples.

- Definitions:

Ais a list of elements or terms, which is completely determined or defined by its elements, and the order in which the elements are listed is irrelevant. We enclose the elements of a set in braces.*set*

Ais a list of elements or terms, which is determined or defined by its elements*sequence*the order in which they are listed.*and*

Anis a list of n elements or terms, which is determined or defined by its elements*ordered n-tuple*the order in which they are listed. We enclose the elements of an ordered tuple in parentheses. An ordered 2-tuple is called an ordered pair, an ordered 3-tuple is called an ordered triple.*and* - Examples:

These sets are the same: {1, 5, 3, 17} and {17, 1, 3, 5}

These sequences are not the same: 1, 5, 3, 17 and 17, 1, 3, 5

These ordered pairs are not the same: (5, 3) and (3, 5)

- Definitions:
- Power Set.

- Definition: If S is a set, then the
of S is the set of all subsets of S.*power set* - Notation: We will use script P to denote power set.
Thus, we will write (S)
for the power set of S.
- Examples:

- Let S = {1, 5}.
Then (S) =
{, {1}, {5}, S},
a set with 4 elements.
- Let S = .
Then (S) = ?
- Let S = {}.
Then (S) = ?
- Let S = {x, y, z}.
Then (S) = ?

- Let S = {1, 5}.
Then (S) =
{, {1}, {5}, S},
a set with 4 elements.

- Definition: If S is a set, then the
- Cartesian Product of Sets.

- Definition: If A and B are sets, then the
of A and B is the set of all ordered pairs (a, b), such that a is in A and b is in B.*cartesian product* - Notation: We will use
to denote the cartesian product, and we will read
A B as A
B.*cross* - A B =
{(a, b) | a A
and b B}
- Examples:

- Let A = {1, 5} and B = {17, 4}.
Then A B =
{(1, 17), (1, 4), (5, 17), (5, 4)}
- Let A = {x, y} and B = {17, 4}.
Then A B = ?
- Let A = {17, 4}.
Then A A = ?
- Let A = {1, 5, 17, 4, -45, 3019, -2} and B = {17, 4, -13, 12}.

Then how many elements are there in the set A B?

- Let A = {1, 5} and B = {17, 4}.
Then A B =
{(1, 17), (1, 4), (5, 17), (5, 4)}

- Definition: If A and B are sets, then the
- Disjoint Sets, Partition of a Set.

- Definition: Two sets are
if they have no elements in common.*disjoint*

Thus, two sets A and B are disjoint A B = . - Definition: Sets A
_{1}, A_{2}, ..., A_{n}are(or pairwise disjoint, or nonoverlapping) if for all i, j in 1, 2, ..., n, A*mutually disjoint*_{i}and A_{j}are disjoint whenever i j.

Thus, A_{1}, A_{2}, ..., A_{n}are mutually disjoint i, j {1, 2, ..., n} i j A B = - Definition: A set of
sets {A*nonempty*_{1}, A_{2}, ..., A_{n}} is aof a set A if:*partition*

- A = A
_{1}A_{2}... A_{n} - A
_{1}, A_{2}, ..., A_{n}are mutually disjoint

*union of mutually disjoint subsets.* - A = A
- Examples:

- A = {5, 17, 4, 3}, A
_{1}= {17}, A_{2}= {5, 3}, A_{3}= {4}

Is {A_{1}, A_{2}, A_{3}} a partition of A? - A = {5, 17, 4, 3}, A
_{1}= {17}, A_{2}= {4, 5, 3}, A_{3}=

Is {A_{1}, A_{2}, A_{3}} a partition of A? - B =

B_{1}= {n | k | n = 2k}, the even integers

B_{2}= {n | k | n = 2k+1}, the odd integers

Is {B_{1}, B_{2}} a partition of B? - S =

C_{1}=^{+}, the positive integers

C_{2}=^{-}, the negative integers

C_{3}= {0}

Is {C_{1}, C_{2}, C_{3}} a partition of S? - N =
^{+}, the positive integers, aka natural numbers {1, 2, 3, ...}

P = {n^{+}| the only positive divisors of n are n and 1}, the prime numbers

C = {n^{+}| there is a positive divisor of n other than n and 1}, the composite numbers

Is {P, C} a partition of N? - P = the xy plane

Q1 = the upper left quadrant

Q2 = the upper right quadrant

Q3 = the lower right quadrant

Q4 = the lower right quadrant

Is {Q1, Q2, Q3, Q4} a partition of P?

- A = {5, 17, 4, 3}, A

- Definition: Two sets are