STEP-BY-STEP LESSON · §1.2
The Language of Sets
Distinguish elements, sets, and ordered pairs.
Before you begin
Variables
In x+y=0, each variable keeps its chosen value throughout the statement. If x=3, only y=−3 satisfies this condition.
x+y=0; x=3 ⇒ y=−3
The condition x>0 depends on the permitted values of x. Specify the domain: integers.
x∈ℤ
Symbols
- ∈
- is an element of
- ⊆
- is a subset of
- ×
- Cartesian product
- ∅ / |A|
- empty set / number of elements of A
Definitions and notation for this topic
Integers
ℤ
Numbers with no fractional part, including zero and negative numbers.
The number 2.5 is not an integer.
Separate glossary example
… −2, −1, 0, 1, 2 …
Positive integers
ℤ⁺
Integers strictly greater than zero.
Zero is not positive.
Positive means greater than zero. The nonnegative integers are the distinct set {0,1,2,…}; state the starting index when using either set.
Separate glossary example
1, 2, 3, …
Natural numbers
ℕ
The symbol ℕ has two common conventions: {1,2,3,…} or {0,1,2,…}. State which one is intended.
Epp avoids the ambiguous term natural numbers here and distinguishes positive from nonnegative integers. Do not infer the lecturer’s zero convention from the symbol alone.
Separate glossary example
1, 2, 3, … or 0, 1, 2, …
Rational numbers
ℚ
Numbers expressible as a/b, where a and b are integers and b ≠ 0.
They can be positive, negative or zero. Only the denominator b must be nonzero. ℚ is a special style of Q; handwritten Q on the board denotes the same set.
Separate glossary example
3/4; −2 = −2/1; 0 = 0/1
Real numbers
ℝ
The numbers on the number line, including rational and irrational numbers.
Real numbers include all rational numbers and irrational numbers. Negative values and zero are allowed. ℝ is a special style of R; it denotes a different set from ℚ.
Separate glossary example
−1/2, 0, √2, −√2, π
Inequality
a<b; a>b
For real numbers, a<b says a is less than b; a>b says a is greater than b.
Read this as: two is less than five.
Separate glossary example
2 < 5
At most / at least
a≤b; a≥b
a≤b means a is at most b; a≥b means a is at least b. Equality is included in both comparisons.
The value 2 is allowed, as is any larger value.
Separate glossary example
n ≥ 2
Absolute value
|x|
The distance from a real number x to zero on the number line.
Distance is nonnegative. These bars mean absolute value for a number, but cardinality in |A| for a set.
Separate glossary example
|−3|=3; |3|=3; |0|=0
Element / membership
∈
The object belongs to the set.
The number 3 is an element of this set.
In 3∈A, the left side is an element and A is a set. A⊆B compares two sets. Neither expression performs the assignment :=.
Separate glossary example
3 ∈ {2, 3, 4}
Not an element of
∉
The object does not belong to the set.
The number 5 is not an element of this set.
Separate glossary example
5 ∉ {2, 3, 4}
Subset
⊆
Every element of A is also an element of B.
Equality is allowed: every set is a subset of itself.
Separate glossary example
{1, 2} ⊆ {1, 2, 3}
Proper subset
⊊
A is a subset of B, and A is different from B.
B has at least one element that is not in A.
Separate glossary example
{1} ⊊ {1, 2}
Set
{a,b}A collection of objects; order and repeated listings do not change a set.
Both sets have exactly the same elements.
Separate glossary example
{1, 2} = {2, 1, 1}
Set-builder notation / such-that bar
{x ∈ D | P(x)}The set of all objects x in D satisfying P(x). Read the bar as “such that.”
Here | means neither divisibility nor absolute value; its meaning depends on the notation.
Separate glossary example
{x∈ℤ | x>0} = {1,2,3,…}
Cardinality
|A|
For a finite set, its number of elements.
Here the vertical bars mean the size of a set.
For infinite sets, cardinality is compared using bijections; it is not a finite element count. See Cardinality and countability. For a number x, |x| is absolute value instead.
Separate glossary example
|{2,3,4}| = 3
Cartesian product
A × B
The set of all ordered pairs (a,b) with a ∈ A and b ∈ B.
The order of the entries in a pair matters.
Separate glossary example
{1} × {2,3} = {(1,2),(1,3)}
Such that
Introduces a condition that the chosen object must satisfy.
There exists an integer x such that x is greater than 2.
Separate glossary example
∃x ∈ ℤ s.t. x > 2
Step by step
Step 1 / 6
A collection of objects
A set specifies which objects are included. Listing order and repetition do not change it.
{2,5}={5,2,2}Compare elements, not how they are listed.
Worked example
Let A={1,3}, B={0,2}. List A×B.
- Choose first coordinate 1 from A. Pair it with each allowed second coordinate: (1,0),(1,2).
- Repeat for first coordinate 3: (3,0),(3,2).
- Therefore A×B={(1,0),(1,2),(3,0),(3,2)}. Each of the two first choices has two second choices, so all four pairs are included.
Optional self-check
Is {2}∈{2,5} true?
Show answer
No. The elements on the right are numbers 2 and 5, not the set {2}.