Chapter 2 - Sets and Venn diagrams

B ⊆ A

If B ⊆ A then every element of B is also in A

Writing in interval notation based on number line

{ 𝑥 ∈ R | -2 ≤ 𝑥 < 4 }

N

{0, 1, 2, 3, 4, 5, 6, 7, ...} is the set of all natural or counting numbers

Z

{0, ±1, ±2, ±3, ±4, ...} is the set of all integers

Z+

{1, 2, 3, 4, 5, 6, 7, ...} is the set of all positive integers

Suppose U = R Copy the Venn diagram and label the sets Q, Z, and N. Shade the region representing Q'. Place these numbers on the Venn diagram: 7, 1/5, 2.8 recurring, -π, 0, square root of 7, -2, -0.35

Q = {-2, -0.35, 0, 1/5, 2.8 recurring, 7} Z = {-2, 0, 7} N = {0, 7} Q' (shaded yellow) = {-π, square root of 7}

interval notation

an expression that uses inequalities to describe subsets of real numbers

a set

is a collection of distinct numbers; each number is called an element or member of the set

Q'

is the set of all irrational numbers, or numbers which cannot be written in rational form

R

is the set of all real numbers, which are all numbers which can be placed on the number line

[ a , b ]

represents the closed interval: { 𝑥 | a ≤ 𝑥 ≤ b }

( a , b ]

represents the interval: { 𝑥 | a < 𝑥 ≤ b }

[ a , b [

represents the interval: { 𝑥 | a ≤ 𝑥 < b }

] a , b ]

represents the interval: { 𝑥 | a < 𝑥 ≤ b }

[ a , b )

represents the interval: { 𝑥 | a ≤ 𝑥 < b }

] a , b [

represents the open interval: { 𝑥 | a < 𝑥 < b }

( a , b )

represents the open interval: { 𝑥 | a < 𝑥 < b }

Q

the set of all rational numbers, or numbers which can be written in the form p/q where p and q are integers, q ≠ 0

A ⊂ B

A is a proper subset of B if every element of A is also an element of B, but A ≠ B

For any set A with complement A':

A ∩ A' = ∅ as A' and A have no common members A ∪ A' = U as all elements of A and A' combined make up U n(A) + n(A') = n(U) provided U is finite

A ⊆ B

Set A is a subset of set B if every element of A is also an element of B. For all 𝑥 ∈ A, we know that 𝑥 ∈ B

A'

The complement of a set A is the set of all elements of U that are not elements of A

A ∩ B

The intersection of two sets A and B is the set of elements that are in both set A and set B

A ∪ B

The union of two sets A and B is the set of elements that are either set A or B.

U

The universal set is the set of all elements we are considering

Example of interval notation

To describe the set of all integers between -3 and 5, we can list the set as {-2, -1, 0, 1, 2, 3, 4} or illustrate the set as points on a number line. Using interval notation: { 𝑥 ∈ Z | -3 < 𝑥 < 5 }

A ∩ B = ∅

Two sets A and B are disjoint or mutually exclusive if they have no elements in common