Chapter 2 · Test 1
Sets, Numbers & Functions for Economics
Set operations, number properties, point sets, functions, and convexity.
What you will learn
- Compute a set intersection
- Compute a complement
- Calculate a convex combination
Worked example
Compute a set intersection: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Compute a set intersection
Let A = {1, 2, 4, 7} and B = {2, 3, 4, 8}. How many elements are in A ∩ B?
A ∩ B = {x : x ∈ A and x ∈ B}In plain English: The intersection of A and B is the set of all x such that x belongs to A and x belongs to B.
Bold blue terms in this symbol guide include definitions. Hover, focus, or tap a term. Each row also shows how to read the notation aloud.
- A, B
- the two setsHow to read: A and B
- x
- a candidate element being tested for membershipHow to read: x
- ∈
- the Set membership: The relation x ∈ A stating that x belongs to set A; x ∉ A states that x does not belong to A. symbol: the object on the left belongs to the set on the rightHow to read: is an element of
- :
- in Set-builder notation: A way to define a set by naming a typical element and stating the condition that it must satisfy, such as {x : P(x)}., introduces the condition that an element must satisfyHow to read: such that
- A ∩ B
- the Set intersection: The set containing exactly the elements that belong to every set being compared. of A and B, containing exactly the elements shared by both setsHow to read: the intersection of A and B
- |A ∩ B|
- the Cardinality: The number of distinct elements in a set; for a finite set A, it is commonly written |A|. of the Set intersection: The set containing exactly the elements that belong to every set being compared., or the number of distinct elements it containsHow to read: the number of elements in the intersection of A and B
Solution
- Compare the elements of A with the elements of B.
- The shared elements are 2 and 4, so A ∩ B = {2, 4}.
- Count the distinct shared elements: |A ∩ B| = 2.
Logic and solving tips
- Intersection means AND: an element must satisfy membership in both sets.
- Treat intersection as an AND test and count each shared element only once.
Where economists use it
A labor economist intersects employment and insurance records to count workers who meet both conditions before estimating coverage gaps or program eligibility.
Key idea: Intersection means AND: an element must satisfy membership in both sets.
Applied case study
Who counts as unemployed in the CPS?
Scenario
Statistical agencies use simultaneous conditions to classify people. BLS generally counts a person as unemployed when the person was not employed, was available for work, and actively searched during the prior four weeks, with a stated exception for temporary layoff.
Problem
In a teaching sample, N = {1,2,3,4,5,6} have no employment, A = {1,2,4,7} actively searched, and V = {1,2,3,4,8} were available. Find N ∩ A ∩ V and its size.
U = N ∩ A ∩ V
In plain English: The unemployment set contains only the people who are simultaneously not employed, actively searching, and available for work.
Worked solution
- First intersect no employment and active search: N ∩ A = {1,2,4}.
- Check those members against availability: 1, 2, and 4 are all in V.
- Therefore U = {1,2,4}.
- Count the members: |U| = 3.
Economic interpretation: Three respondents satisfy all three conditions. Meeting only one or two conditions is insufficient under the simplified rule.
Source: U.S. Bureau of Labor Statistics: Current Population Survey Concepts and Definitions The classification conditions and temporary-layoff exception are sourced. The respondent IDs and memberships are teaching data.
Reasoning habits that help
- Intersection means AND: an element must satisfy membership in both sets.
- A set complement is meaningless until the universal set has been specified.
- Keep λ symbolic to construct the whole segment, substitute a stated λ for one point, or solve for λ and verify 0 ≤ λ ≤ 1 to test membership.
Where economists use these methods
- A labor economist intersects employment and insurance records to count workers who meet both conditions before estimating coverage gaps or program eligibility.
- A policy agency takes the complement of benefit recipients to identify eligible households that a program has not reached.
- A portfolio manager uses weights such as 40% bonds and 60% equities to calculate the return and risk of a feasible mixed portfolio and compare it with holding either asset alone.
Chapter vocabulary
Study 100 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
SetA = {1, 2, 3}
A collection of objects treated as one mathematical object.
How to read: A is the set containing one, two, and three
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 11Elementx ∈ A
An individual object belonging to a set.
How to read: x is an element of A
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 11Definition by enumeration
A way to define a set by listing all of its elements inside braces.
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed pages 11-12Definition by propertyA = {x : P(x)}
A way to define a set by stating the condition that exactly its elements satisfy.
How to read: A is the set of x such that x has property P
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed pages 11-12Set membershipx ∈ A; x ∉ A
The relation that states whether an object belongs to a set.
How to read: x is in A; x is not in A
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 12Cardinality of a set|A|
The number of distinct elements in a set.
How to read: the number of elements in A
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 23SubsetA ⊆ B
A set A is a subset of B when every element of A is also an element of B.
How to read: A is a subset of B
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 13Proper subsetA ⊂ B
A subset of another set that is not equal to that set.
How to read: A is a proper subset of B
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 13Equality of setsA = B
Two sets are equal when they contain exactly the same elements.
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 13Empty set∅
The unique set containing no elements.
How to read: the empty set
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 15Singleton
A set containing exactly one element.
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 15Disjoint setsA ∩ B = ∅
Sets that have no elements in common.
How to read: A intersection B is empty
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 16IntersectionA ∩ B
The set of elements belonging to every set being compared.
How to read: A intersection B
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 15UnionA ∪ B
The set of elements belonging to at least one of the sets being combined.
How to read: A union B
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 16Set differenceA - B
The elements of one set that are not elements of another set.
How to read: A minus B
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 18Universal setU
The full collection of elements under consideration in a particular problem.
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 14ComplementA^c = U - A
The elements in the universal set that are not in the specified set.
How to read: A complement equals U minus A
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 17Power setP(A)
The set containing every subset of a given set, including the empty set and the set itself.
How to read: the power set of A
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 21Partition
A collection of nonempty, pairwise disjoint subsets whose union is the original set.
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 19Venn diagram
A diagram that represents sets as regions so membership, overlap, and inclusion can be seen.
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 15Natural numbersZ_+ = {1, 2, 3, ...}
The positive whole numbers used for counting.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 23One-to-one correspondence
A pairing in which each element of either set is matched with exactly one element of the other.
Example: Counting pairs each object with one natural number.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 23Closure under an operation
A set is closed under an operation when applying it to members of the set always produces another member of the set.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 24IntegersZ = {..., -2, -1, 0, 1, 2, ...}
The positive and negative whole numbers together with zero.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 24Rational numbersQ = {a/b : a, b ∈ Z, b ≠ 0}
Numbers expressible as a ratio of two integers with a nonzero denominator.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 25Irrational numbers
Real numbers that cannot be written as a ratio of two integers.
Example: √2 and π are irrational.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed pages 25-26Real numbersℝ
The set containing every rational and irrational number. Each real number corresponds to exactly one point on an unbroken number line.
How to read: the real numbers
Example: −5, 0, 1/2, 0.75, √2, and π are all in ℝ. One real coordinate, such as 5, identifies one point on the line.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 26Real lineℝ
The one-dimensional geometric picture of ℝ: an unbroken line on which every real number has exactly one location and every point represents exactly one real number.
How to read: the real line
Example: The point 5 lies in ℝ and needs one coordinate. The point (5, 3) needs two coordinates, so it lies in ℝ² rather than on the real line.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 26Completeness property
Every nonempty set of real numbers that is bounded above has a least upper bound in the real numbers.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed pages 30, 33Reciprocal
For a nonzero number a, the number 1/a whose product with a equals one.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 27Nonnegative real numbersℝ₊ = {x ∈ ℝ : x ≥ 0}
The part of ℝ containing zero and every positive real number. Nonnegative means greater than or equal to zero, so zero is included and negative values are excluded.
How to read: R plus is the set of real x such that x is greater than or equal to zero
Example: 0, 2.5, and √2 belong to ℝ₊, while −1 does not. ℝ₊ has one nonnegative coordinate; ℝ₊² has two, such as the economic pair (labor, output) = (4, 10).
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 28Strict inequality
An ordering relation that excludes equality, such as x > y or x < y.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 28Weak inequality
An ordering relation that permits equality, such as x ≥ y or x ≤ y.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 28Dimensions of an economic variable
The units in which an economic quantity is measured, such as dollars, hours, or dollars per unit.
Example: Marginal cost may be measured in dollars per additional unit of output.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 29Pure number
A dimensionless number formed when measurement units cancel or no units apply.
Example: A rate of return measured as profit divided by revenue is a pure number.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 30Relative differenceX - Y = {x ∈ X : x ∉ Y}
The elements of one set that are not in another set. In this textbook, relative difference is the same operation also called set difference.
How to read: X minus Y
Textbook reference: Chapter 2, Section 2.1 Sets and Subsets, printed page 18Supremum
The least upper bound of a set: the smallest real number that is at least as large as every member of the set.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed pages 30, 59Infimum
The greatest lower bound of a set: the largest real number that is no greater than every member of the set.
Textbook reference: Chapter 2, Section 2.2 Numbers, printed page 59Interior numbers (textbook review label)
The chapter review lists this phrase but does not define it in the chapter. It appears to be an editorial error. The defined geometric concept to study is interior point.
Textbook reference: Chapter 2, Section Chapter Review terminology note, printed page 58Coordinate system
A set of reference axes used to identify a point by an ordered list of coordinates.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 31Ordered pair(x, y)
A pair (x, y) whose position matters, used to identify a point in two-dimensional space.
How to read: the point x comma y
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 31Cartesian productX × Y
The set of all ordered pairs formed by taking one element from each of two sets.
How to read: X cross Y
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 32n-dimensional real spaceℝⁿ
The set of all ordered lists of n real coordinates. The exponent tells you how many coordinates identify one point: ℝ uses one, ℝ² uses two, ℝ³ uses three, and ℝⁿ uses n.
How to read: R to the n, or n-dimensional real space
Example: 5 ∈ ℝ is a point on a line; (5, 3) ∈ ℝ² is a point on a plane; and (5, 3, −1) ∈ ℝ³ is a point in three-dimensional space. An economic pair such as (price, quantity) belongs to ℝ².
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 32Point set
A subset of ℝⁿ whose elements are points represented by ordered n-tuples.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 32Interval
A set of real numbers containing every real number between any two of its members.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 33Closed interval[a, b]
An interval containing both endpoints.
How to read: the closed interval from a to b
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 33Open interval(a, b)
An interval containing neither endpoint.
How to read: the open interval from a to b
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 33Half-open interval[a, b) or (a, b]
An interval containing one endpoint and excluding the other.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 33Interior point
A point surrounded by some neighborhood that lies entirely within the set.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 33Boundary point
A point whose every neighborhood contains both points in the set and points outside it.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed pages 33, 36Convex combinationλx + (1 - λ)x', 0 ≤ λ ≤ 1
A weighted average of points whose nonnegative weights sum to one.
How to read: lambda times x plus one minus lambda times x prime
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed pages 34, 37Euclidean distanced(a, b) = √Σ_i(a_i - b_i)^2
The straight-line distance between two points, found from the square root of the sum of squared coordinate differences.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed pages 34-35Epsilon-neighborhoodN_ε(x_0) = {x ∈ ℝⁿ : d(x_0, x) < ε}
The set of points lying less than a positive distance epsilon from a specified point.
How to read: the epsilon neighborhood of x naught
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed pages 35-36Open set
A set in which every point has some neighborhood contained entirely in the set.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 36Closed set
A set whose complement is open, equivalently a set that contains all of its boundary points.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 37Bounded set
A set whose points all fit inside some finite-radius neighborhood.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 37Compact set
In finite-dimensional real space, a set that is both closed and bounded.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 33Convex set
A set containing the entire line segment between every pair of its points.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 38Strictly convex set
A convex set in which every interior convex combination of two distinct points is an interior point of the set.
Textbook reference: Chapter 2, Section 2.3 Some Properties of Point Sets in ℝⁿ, printed page 39Functionf: X → Y
A rule assigning each element of a domain exactly one element of a codomain.
How to read: f maps X to Y
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 41Mapping
Another name for a function, emphasizing the assignment from a domain to a codomain.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 42Domain
The set of permitted inputs to a function.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 41Codomain
The set in which a function's outputs are declared to lie, whether or not every member is attained.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 41Range
The set of output values actually produced by a function over its domain.
Textbook reference: Chapter 2, Section 2.4 Functions, printed pages 41-42Image
The output f(x) assigned to a particular input x.
Textbook reference: Chapter 2, Section 2.4 Functions, printed pages 41-42Image setf(X) = {f(x) : x ∈ X}
The set of all images produced from a specified set of inputs.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 42Real-valued function
A function whose outputs are real numbers.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 42Independent variable
An input variable whose value is supplied to a functional relationship.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 45Dependent variable
An output variable whose value is determined by the function's inputs.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 45One-to-one function
A function in which distinct inputs always produce distinct outputs.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 42Onto function
A function whose range equals its codomain.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 42Inverse functionf^{-1}(y)
A function that reverses a one-to-one function by mapping each output back to its unique input.
How to read: f inverse of y
Textbook reference: Chapter 2, Section 2.4 Functions, printed pages 42-43Composite mapping(g ∘ f)(x) = g(f(x))
A function formed by applying one function and then applying another to the result.
How to read: g composed with f of x
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 43Linear functiony = ax + b
A function whose graph is a straight line and whose slope is constant.
Textbook reference: Chapter 2, Section 2.4 Functions, printed pages 44-45SlopeΔy/Δx
The change in a line's vertical coordinate divided by the corresponding change in its horizontal coordinate.
How to read: change in y divided by change in x
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 45Slope coefficient
The constant multiplying x in a linear function, equal to the line's slope.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 45Intercept term
The constant term in a linear function, equal to y when x is zero.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 45Implicit functionF(x, y) = 0
A relationship among variables written as an equation without isolating one variable as the output.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 45Quadratic functiony = ax^2 + bx + c, a ≠ 0
A polynomial function whose highest power is two, producing a parabolic graph.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 48Rectangular hyperbolaxy = a or y = a/x
A curve defined by a constant product of two variables.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 48Exponenta^b
A number indicating the power to which a base is raised.
How to read: a to the power b
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 49Power functiony = ax^b
A function in which the input is raised to a fixed exponent.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 49Exponential functiony = ab^x
A function in which the input appears in the exponent of a fixed positive base.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 49Base
The fixed number raised to a variable or fixed exponent in an exponential or logarithmic expression.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 49Logarithmic functiony = log_b(x) means b^y = x
The inverse of an exponential function, returning the exponent needed to obtain an input from a chosen base.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 50Natural logarithmln x = log_e(x)
The logarithm with base e, where e is approximately 2.718.
How to read: the natural log of x
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 50Concave functionf(λx + (1 - λ)y) ≥ λf(x) + (1 - λ)f(y)
A function whose value at every convex combination of inputs is at least the same convex combination of their function values.
Textbook reference: Chapter 2, Section 2.4 Functions, printed pages 51-52Strictly concave function
A concave function for which the defining inequality is strict between distinct inputs and interior weights.
Textbook reference: Chapter 2, Section 2.4 Functions, printed pages 51-52Convex functionf(λx + (1 - λ)y) ≤ λf(x) + (1 - λ)f(y)
A function whose value at every convex combination of inputs is no greater than the same convex combination of their function values.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 52Strictly convex function
A convex function for which the defining inequality is strict between distinct inputs and interior weights.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 52Level setL_c = {x : f(x) = c}
The set of inputs that produce the same specified function value.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 53Cobb-Douglas functionf(x_1, x_2) = A x_1^α x_2^β
A multiplicative power function commonly used to represent production or preferences.
Textbook reference: Chapter 2, Section 2.4 Functions, printed pages 53-54Indifference curve
A consumer-theory level set containing bundles that provide the same utility.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 54Isoquant
A production-function level set containing input combinations that produce the same output.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 54Better setB(x^0) = {x : f(x) ≥ f(x^0)}
For a reference input, the set of inputs producing at least as large a function value.
Textbook reference: Chapter 2, Section 2.4 Functions, printed pages 54-55Quasiconcave function
A function for which every upper contour set, or better set, is convex.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 55Worse setW(x^0) = {x : f(x) ≤ f(x^0)}
For a reference input, the set of inputs producing no greater a function value.
Textbook reference: Chapter 2, Section 2.4 Functions, printed pages 55-56Quasiconvex function
A function for which every lower contour set, or worse set, is convex.
Textbook reference: Chapter 2, Section 2.4 Functions, printed page 56Necessary condition
A condition that must hold whenever a stated result is true, although it may not be enough by itself to guarantee the result.
Textbook reference: Chapter 2, Section Chapter Review, printed page 59Sufficient condition
A condition that guarantees a stated result, although the result may also occur without that condition.
Textbook reference: Chapter 2, Section Chapter Review, printed page 59End-of-chapter problems
Practice all 9 Chapter Review Exercises
The index covers every numbered Review Exercise in this chapter of Mathematics for Economics, 3rd edition. Each reference opens the textbook exercise itself, including every listed subpart, with MFE hints, a symbol guide, a worked explanation, progress tracking, and similar-problem practice.
Practice end-of-chapter problems