Simple Statements and Sets
- Tyler Buffone
- Dec 23, 2025
- 4 min read
Math is picky about language for a reason.
A lot of mathematics is about patterns, but the real superpower is being able to talk about patterns precisely. Proofs, definitions, and even good problem-solving depend on sentences where we can tell what they mean and whether they are correct.
So we start with a simple question:
When does a sentence count as a mathematical statement?
What is a statement?
A statement (often called a proposition) is a sentence that makes a definite claim, in a way that it makes sense to label it as either:
True, or
False
and not both.
Examples of statements
These are statements because each one has a clear truth value:
8 + 5 = 13 (true)
17 is an even number (false)
Calgary is in Alberta (true)
√49 = 8 (false)
The key is not whether the statement is about math. The key is whether the sentence is a clear assertion with a truth value.
Non-statements: sentences with no truth value
Many English sentences are perfectly meaningful, but they are not statements in the logical sense because they don’t land on “true” or “false.”
Common non-statement types
Commands (imperatives)
“Solve this equation.” This is an instruction, not a claim.
Questions (interrogatives)
“What is 9 squared?” A question has no truth value until it is answered.
Exclamations
“That’s amazing!” This expresses emotion, not a claim with a definite truth value.
So in logic, the most important sentence type is the declarative type: it asserts something that can be judged.
Open sentences (sentences with variables)
Sometimes a sentence looks like a statement, but it isn’t, because it depends on an unspecified variable.
Example:
x² + 1 = 5
Is that true or false? It depends on what x is.
If x = 2, then x² + 1 = 4 + 1 = 5 (true)
If x = 0, then x² + 1 = 0 + 1 = 1 (false)
So this is not a statement yet. It becomes a statement only after you do one of the following:
Choose a value of the variable
“When x = 2, x² + 1 = 5.”
Specify a domain and quantify
“There exists a real number x such that x² + 1 = 5.”
“For every real number x, x² + 1 = 5.” (false)
Quantifiers (for all / there exists) will be a major topic later because they are what turn “open sentences” into full statements.
A quick note on self-referential sentences
Some sentences cause trouble because they refer to themselves in a way that breaks the true/false system. A famous example is the “liar”-type construction:
“This sentence is false.”
In a typical introductory logic setting, the main takeaway is simple: logic needs well-formed statements, and some self-referential forms are not treated as ordinary statements because they don’t behave like “true” or “false” claims.
(You do not need deep philosophy here. For most math courses, it’s just a warning label: not every grammatical sentence belongs in our logical toolbox.)
Symbols that appear inside statements
In math, many statements are built using relation symbols that compare objects.
Equality
= means “is equal to”
It connects two expressions that represent the same object/value.
Example:
3(4) = 12
Inequalities
> means “greater than”
< means “less than”
≥ means “greater than or equal to”
≤ means “less than or equal to”
Example statements:
-2 < 5 (true)
10 ≤ 7 (false)
Negations of relations
Math also uses symbols that negate relations:
≠ means “not equal”
≮ means “not less than” (equivalently, “greater than or equal to”)
≯ means “not greater than” (equivalently, “less than or equal to”)
Example:
Saying a ≠ b is a statement claiming the two expressions do not represent the same object.
Sets: where membership becomes the statement
Many mathematical claims are not about “greater than” or “equals.” They are about membership.
A set is a well-defined collection of distinct objects (called elements), where for any object you mention, it makes sense to ask:
Is it in the set, or not?
If that question is not decidable from the description, then the “collection” is not a set in the mathematical sense.
Membership symbols
x ∈ A means “x is an element of A”
x ∉ A means “x is not an element of A”
Example: Let ℤ be the set of integers.
3 ∈ ℤ (true)
1/2 ∈ ℤ (false)
Notice: these are full statements. They have truth values.
When are two sets equal?
Two sets are equal if they contain exactly the same elements.
That means order does not matter, and repetition does not matter:
{1, 2, 3} = {3, 2, 1}
{1, 1, 2, 2, 3} = {1, 2, 3}
So when you “define” a set, you are really defining which objects belong to it.
Two ways to describe a set
A) Roster form (listing)
You can list elements inside braces:
{ -1, 0, 1 }
Sometimes we use dots to signal a pattern:
{1, 2, 3, …, 50} means the integers from 1 to 50
{1, 2, 3, …} means the positive integers
B) Rule form (set-builder style, conceptually)
Instead of listing, you describe a rule that lets you decide membership.
Example idea:
“The set of all even integers.”
The important point is not the notation, it’s the logic: the description must give a clear test for membership.
Sets can contain sets (and this matters)
A set’s elements do not have to be numbers. They can themselves be sets.
Example:
{{1, 3}, {5, 7}} is a set whose elements are two smaller sets.
This leads to a classic trap:
{a, b} has two elements: a and b
{{a, b}} has one element: the set {a, b}
So {{a, b}} ≠ {a, b}.
Mini practice (with answers)
Decide whether each is a statement. If it is, label it true or false.
(a) “7 is prime.”
(b) “What is 7 squared?”
(c) “x + 4 = 9”
(d) “Toronto is in British Columbia.”
Let A = {1, 2, 3} and B = {3, 2, 1, 1}. Is A = B?
Let S = {∅, 1, 2}. Which are true?
(a) ∅ ⊆ S
(b) ∅ ∈ S
Answers
(a) statement, true. (b) not a statement (question). (c) not a statement (open sentence). (d) statement, false.
Yes, A = B (same elements; repetition does not add new elements).
Both are true: ∅ is a subset of every set, and here ∅ is also listed as an element of S.
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