MAT148H1 Chap.1 Sets, Interval Notation and Quantifiers
Sets, Interval Notation and Quantifiers
Sets as the vocabulary of proof
A set is a collection of well-defined, distinct objects. The symbol ∈ records membership, so 2 ∈ {1, 2, 3}, and A ⊆ B says that every element of A also belongs to B. Two sets are equal exactly when each is contained in the other, which is why most set equalities in this course are proved in two halves.
Small sets can be listed inside braces, yet the sets that matter in calculus are infinite, so they are described by a rule. Set-builder notation {x ∈ S : P(x)} names the elements x of a universe S for which the condition P(x) holds. The standard universes are ℕ, ℤ, ℚ and ℝ.
Authors disagree on whether 0 belongs to ℕ, so stating a convention avoids confusion later.
Intervals and their endpoints
Intervals are the connected pieces of the real line that appear in almost every later definition.
For real a < b, the open interval (a, b) contains every x with a < x < b, the closed interval [a, b] adds both endpoints, and the forms [a, b) and (a, b], which keep exactly one endpoint, are called half-open. Unbounded intervals such as (−∞, a) or (a, ∞) always use a round bracket beside the infinity symbol, because ∞ is a symbol for unboundedness rather than a real number; the whole line is ℝ = (−∞, ∞).
On a sketch, a filled dot means the endpoint is included and a hollow dot means it is excluded. Converting between notations is routine: the inequality −1 ≤ x < 4 becomes [−1, 4), and x > 7 becomes (7, ∞).
Getting this bracket right is not cosmetic: whether a single boundary point belongs to a set can change whether a function is defined there or whether a supremum is attained.
Universal and existential quantifiers
The quantifier ∀ is read for all and ∃ is read there exists. A statement ∀x ∈ S, P(x) asserts P for every element of S, so a single element that fails is enough to make it false.
A statement ∃x ∈ S, P(x) asserts that at least one witness exists, so producing one explicit witness proves it. Inside a set-builder rule, a quantifier is checked separately for each candidate x: fix x, then ask whether the quantified sentence about that x is true.
For instance, the set of reals x with x < 3 + ε for all ε > 0 turns out to be (−∞, 3], while the set of reals x for which some ε > 0 gives x < 3 − ε is (−∞, 3). One quantifier keeps the endpoint and the other loses it.
Order and negation of quantifiers
Quantifiers of the same kind commute, but a mixed pair does not.
In ∀x ∃y, the value of y is chosen after x and may depend on it; in ∃y ∀x, a single y must serve every x at once. Over the integers, every x has some y with x + y = 0, yet no single y works for all x, so swapping the order turns a true statement into a false one. Negation follows a mechanical rule: ∀ becomes ∃, ∃ becomes ∀, and the final condition is replaced by its opposite.
Practising these translations, from symbols to plain English and back, is the fastest way to prepare for the limit definitions, where three quantifiers appear in one line and their order carries the entire meaning.
What this chapter covers
- 01
Sets, membership and subsets
- 02
Set-builder notation
- 03
Open, closed and half-open intervals
- 04
Universal and existential quantifiers
- 05
Quantified conditions inside set definitions
- 06
Order of quantifiers
- 07
Negating quantified statements
Worked example · free
Converting a quantified set into interval notation
- 2Claim G = (−∞, 3]. To show (−∞, 3] ⊆ G, take x ≤ 3 and any ε > 0; then x ≤ 3 < 3 + ε, so x satisfies the rule.
- 2To show G ⊆ (−∞, 3], argue by contrapositive: take x > 3 and choose ε = (x − 3)/2, which is positive.
- 2Compute 3 + ε = (x + 3)/2, which is less than x because x > 3. This one ε breaks the universal condition, so x ∉ G, and the two inclusions give G = (−∞, 3].
Key terms
- Set-Builder Notation
- A way of naming a set by the property its members share, written {x ∈ S : P(x)}. It describes infinite sets of real numbers that cannot be listed element by element.
- Half-Open Interval
- An interval of real numbers that contains exactly one of its two endpoints, such as [a, b) or (a, b]. It sits between the open interval (a, b) and the closed interval [a, b].
- Universal Quantifier
- The symbol ∀, read for all, asserting that a property holds for every element of a stated set. One element where the property fails is enough to make the whole statement false.
- Existential Quantifier
- The symbol ∃, read there exists, asserting that at least one element of a stated set has the property. Exhibiting a single explicit witness is enough to prove such a statement.
- Witness
- A specific object produced to show that an existential statement is true. In nested statements the witness may depend on variables chosen earlier, such as an ε built from a fixed x.
Sets, Interval Notation and Quantifiers FAQ
What is the difference between (a, b) and [a, b] in interval notation?
Round brackets exclude the endpoint and square brackets include it. So (2, 5) contains 2.5 but neither 2 nor 5, while [2, 5] contains both. On a number line this matches hollow versus filled dots, and infinity always takes a round bracket.
Why does the order of ∀ and ∃ matter?
With ∀x ∃y, each x may receive its own y, so y can depend on x. With ∃y ∀x, one fixed y must serve every x simultaneously, a much stronger demand. Swapping them can turn a true statement into a false one.
How do I negate a statement with several quantifiers?
Work left to right. Change every ∀ into ∃ and every ∃ into ∀, keep the same variables and sets, and replace the final condition by its opposite. Then translate back into English to check that the result really says the reverse.
How do I prove two sets are equal?
Show each set is contained in the other. Take an arbitrary element of the first set and verify it satisfies the rule of the second, then repeat in the other direction. For interval answers, include an argument for every excluded point.
Exam move
Spend ten minutes a day translating: take three symbolic statements from your notes, rewrite each in plain English without variables, then negate it in symbols. For every set-builder rule you meet, draw the number line, mark the boundary point, and decide with a written check whether it belongs. Once a week, swap the order of quantifiers in one true statement and prove the new version true or false.
Keep a page listing each witness choice you used, like half the distance to the boundary.
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