## Sets Math – Symbols Used in Set Theory

A set is defined by a collection of objects. The objects of a particular set are its elements. Set theory is the study of such sets, and the relation that connects the sets to its elements. Set Theory was first created by Sir Georg Ferdinand Ludwig Philipp Cantor who was a German mathematician. The history and creation of set theory is quite different from the history of other areas of mathematics. It was created to be able to talk about collections of objects that represent a particular set. Set theory has turned out to be an extremely important tool for defining some of the most complex and significant mathematical structures.

- • The collection of all green bottles.

- • The collection of negative numbers.

- • The collection of people born before 1995.

- • The collection of greatest football players.

All of the above collections are sets. However, the collection of greatest football players is not well-defined. Usually, we restrict our focus to just well-defined sets.

- • Descriptive form

- • Roster(listing) form

- • Set builder form

For example: Define/represent a set of odd natural numbers that lies between the interval of 2 to 12 in descriptive, roster and set builder form.

Example: The set of all odd counting numbers between 2 and 12.

Roster (listing) form

Example: {3, 5, 7, 9, 11}

Set builder form

Example: {x | x is a natural number, x is odd, and 2< x < 12}

Note:

- 1. Use curly braces to represent sets.

- 2. Use commas to separate set elements from each other.

- 3. The variable in the set–builder notation doesn’t necessarily have to be x.

- 4. The ellipses (. . .) are used to indicate a continuation of a pattern/ series established before the ellipses i.e. {1, 2, 3, 4 . . . 100}.

- 5. The symbol | is called “such that”.

Set Membership

- • N denotes Natural or Counting numbers: {1, 2, 3 . .}

- • W denotes Whole Numbers: {0, 1, 2, 3 . . .}

- • I denote Integers: {. ., -5, -4, -3, -2, -1, 0, 1, 2, 3 . .}

- • Q denotes Rational numbers: {p/q | p, q ∈ I, q 6= 0}.

- • R denotes Real Numbers: {x | x is a number that can be written as a decimal}.

- • Irrational numbers: {x | x is a real number and it cannot be written as a quotient of integers}

- • Examples are: π, √ 3, and
^{3}√ 9.

- • ∅ represents Empty Set: {}, the set that contains nothing.

- • U represents Universal Set: the set of all objects currently under discussion.

NOTE: Any rational number can be written either as a terminating decimal (e.g 0.5, 0.333, or 0.8578966) or as a repeating decimal (e.g 0.333 or 123.392545).

Is ∅∈ {a, b, c}?

Is ∅∈ {∅, {∅}}?

Is ∅∈ {{∅}}?

Is 1/3 ∈ {x | x = 1/p, p ∈ N}?

If A = {5, 7, 9, 11, 13}

B = {5, 10, 15 . . . 100}

C = {3, 5, 7, 9, . . .}

D = {1, 3, 3, 1, 2}

P = {x | x is odd, and x < 12}

n (A) =?

n (B) =?

n(C) =?

n (D) =?

n (P) =?

- i. Every element of P is an element of Q,

- ii. Every element of Q is an element of P. In other words, set P and Q are equal if and only if they contain exactly the same elements

{x, y, z} = {z, x, y} = {z, y, y, y, x, x,}

{3} = {x | x ∈ N and 1 < x < 5}?

{x | x ∈ N and x < 0} = {y | y ∈ Q and y is irrational}

Question

What is: P’, the complement of P?

What is: R’, the complement of R?

What is: Q’, the complement of Q?

What is: ∅’, the complement of ∅?

- i. P ⊆ Q and

- ii. Q ⊆ P

Proper Subset: P ⊂ Q if P ⊆ Q and P = Q

Whether or Not a Subset?

Is the set on the left a subset of the set on the right?

- a. {a, b, c} {a, c, d, f}

- b. {a, b, c} {c, a, b}

- c. {a, b, c} {a, b, c}

- d. {a} {a, b, c}

- e. {a, c} {a, b, c, d}

- f. {a, c} {a, b, d, e, f}

- g. X X

- h. ∅ {a, b, c}

- i. ∅ ∅

Points to remember:

- • Any set is a subset of itself and also a subset of the universal set.

- • The empty set is a subset of all the sets as well as itself.

Questions:

Is the set on the left equal to, a proper subset of, or not a subset of the set on the right?

{1, 2, 3} I

{a, b} {a}

{a} {a, b}

{a, b, c} {a, d, e, g}

{a, b, c} {a, a, c, b, c}

{∅} {a, b, c}

{∅} {}

S (∅) =

S ({a}) =

S ({a, b}) =

S ({a, b, c}) =

^{ n}.

Set Union

X ∪ Y = {x|x ∈ X or x ∈ Y}. Therefore, for an object to be in X ∪ Y, it must be a member of either X or Y. The total shaded area covered by X and Y falls under XUY.

Also, X ∈ Y means x ∈ Y’. Thus, X − Y = {x|x∈ X and x ∈ Y’} = X ∩ Y’

{a, b, c}∪{b, f, g} =

{a, b, c}∪{a, b, c} =

{ 5, 2, 3, 1, 4} − {2, 4, 6} =

{ 1, 3, 5} {2, 6, 4} − {1, 2, 3, 4, 5} =

U = {1, 2, 3, 4, 5, 6, 9}

P = {1, 2, 3, 4}

Q = {2, 4, 6}

R = {1, 3, 6, 9}

P ∪ Q =

P ∩ Q =

P ∩ U =

P ∪ U=

P’ =

P’ ∩ Q =

P’ ∪ Q =

P ∪ Q ∪ R =

P ∩ Q ∩ R =

P’ ∪ Q’ =

P’ ∩ Q’ =

P ∩ (Q∪R) =

(P’∪R) ∩ Q =

P − Q =

Q − P =

(P − Q) ∪ R’ =

Fill up the Venn diagram to represent U, A, and B

Shade the area in the Diagram

- a. (A ∩ B)’
- b. A’ ∪ B’

Write whether the Following Sets are Disjoint?

{c, b, a} and {d, e, f, g} …

{a, b, c} and {a, b, c} …

{a, b, c} and {a, b, z} …

{a, b, c} and {x, y, z} …

{a, b, c} and ∅ …

For any A, A and ∅ …

For any A, A and A’ …

De Morgan’s Laws

For any sets P and Q,

(P ∩ Q)’ = P’ ∪ Q’

(P ∪ Q)’ = P’ ∩ Q’