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Quizzes > Mathematics > Advanced Math

Are You a Set Theory & Logic Pro? Take the Quiz!

Ready for a challenging logic and sets quiz? Prove your discrete math skills!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art circles squares overlapping on golden yellow background representing sets subsets and logic concepts

Are you ready for the ultimate set theory quiz? Test your grasp of sets, subsets, and logical proofs in this free assessment designed for math enthusiasts. Whether you're brushing up with a set operations quiz, diving into a logic quiz to sharpen your thinking, or exploring a discrete math quiz, this challenge will strengthen your skills. You'll tackle union, intersection, and complement puzzles, explore proof strategies in a logic and sets quiz, and try math and logic questions alongside a reasoning quiz . Ready to prove you've mastered set theory? Click "Start" and take on the challenge now!

What is the union of sets A = {1, 2, 3} and B = {3, 4, 5}?
{1, 2, 3, 4, 5}
{3}
{1, 2, 4, 5}
{}
The union of two sets A and B is the set of all elements that belong to A or B (or both). In this case, elements 1, 2, and 3 are in A, and elements 3, 4, and 5 are in B, so the union is {1, 2, 3, 4, 5}. Note that duplicates are only listed once. More details can be found here.
What is the intersection of sets X = {a, b, c} and Y = {b, c, d}?
{b, c}
{a, d}
{a, b, c, d}
{}
The intersection of two sets contains only those elements that appear in both sets. Here, b and c are the only common elements of X and Y, giving {b, c}. Elements not in both sets are excluded. See more at this page.
If A = {1, 2, 3}, which of the following is a subset of A?
{2, 3}
{1, 4}
{0}
{1, 2, 3, 4}
A subset is any set whose elements all belong to another set. The set {2, 3} contains only elements found in A, so it is a subset. {1, 4} fails because 4 is not in A. For more, refer to Subset definition.
What does the empty set represent?
A set with no elements
A set with infinitely many elements
A set with one element
A set of null numbers
The empty set, denoted ?, contains no elements by definition. It is unique and is a subset of every set. It is neither infinite nor single?element. More info at Empty set.
What is the cardinality of the set {apple, banana, cherry}?
3
2
1
0
Cardinality measures the number of elements in a set. Since there are three fruits listed, the cardinality is 3. It counts each distinct element once. See Cardinality for details.
If U is the universal set and A is a subset of U, what is the complement of A?
All elements in U not in A
All elements in A and U
The union of A and U
The intersection of A and U
The complement of A in U is defined as all elements of the universal set U that are not in A. It excludes exactly those in A. It is neither the union nor intersection. Learn more here.
In set-builder notation, what does { x | x > 0 } represent?
The set of all positive numbers
The set of all negative numbers
The set of natural numbers
The empty set
Set-builder notation { x | condition } describes all x satisfying the given condition. Here x > 0 picks out all positive numbers. It does not restrict to integers. See this explanation.
What is the power set P(A) of a set A?
The set of all subsets of A
The union of A with itself
The complement of A
An infinite set regardless of A
The power set P(A) is defined as the collection of every possible subset of A, including the empty set and A itself. Its cardinality is 2^|A| for finite A. It is not a union or complement. More at Power set.
Which of the following is one of De Morgans laws for sets?
(A ? B)? = A? ? B?
(A ? B)? = A ? B
A ? (B?) = (A ? B)?
A ? (B?) = (A ? B)?
De Morgans first law states that the complement of a union is the intersection of complements: (A ? B)? = A? ? B?. The other form covers intersection complement. Incorrect options mix operations. See De Morgans laws.
What is A \ B when A = {1, 2, 3, 4} and B = {3, 4, 5}?
{1, 2}
{3, 4}
{1, 2, 3, 4, 5}
{5}
A \ B (the set difference) contains elements of A not in B. Removing 3 and 4 from A leaves {1, 2}. It is different from intersection and union. More at Set difference.
If f is a function viewed as a set of ordered pairs, which must always be true?
Each first element appears in exactly one ordered pair
Each second element appears exactly once
Pairs may have duplicate first elements
Pairs may have duplicate second elements but not first
A function as a set of ordered pairs requires that each input (first element) maps to exactly one output. Outputs can repeat for different inputs. This ensures well-defined mapping. See Functions as sets.
What is the Cartesian product A B for A = {1, 2} and B = {a, b}?
{(1,a), (1,b), (2,a), (2,b)}
{(a,1), (b,1), (a,2), (b,2)}
{1,a,2,b}
{(1,1), (2,2), (a,a), (b,b)}
The Cartesian product A B is the set of ordered pairs where the first element is from A and the second from B. That yields four pairs. Order matters. Details at Cartesian product.
What is the symmetric difference A ? B?
(A \ B) ? (B \ A)
A ? B
A ? B
A? ? B?
The symmetric difference A ? B consists of elements in exactly one of the sets, which is (A \ B) ? (B \ A). It excludes the intersection. This operation is commutative. See Symmetric difference.
Which of these sets is countably infinite?
The set of all integers
The set of all real numbers
The empty set
The power set of the naturals
A set is countably infinite if its elements can be matched bijectively with natural numbers. The integers ? are countable. The reals and power set of naturals are uncountable, and the empty set is finite. See Countable set.
A function f: A ? B is injective if which condition holds?
Distinct inputs map to distinct outputs
Every element of B has a preimage
Two inputs can share an output
It is both one-to-one and onto
Injectivity means no two different inputs map to the same output. This is also called one-to-one. Surjectivity refers to covering all outputs. More at Injective function.
In Venn diagrams, how many distinct regions are formed by three overlapping sets?
8
7
6
9
Three overlapping sets partition the plane into 2 = 8 distinct regions corresponding to all inclusion/exclusion combinations. Each region represents a different membership pattern. Understanding these helps with set identities. See Venn diagram.
What does Cantors theorem state about the power set of any set A?
There is no bijection between A and its power set P(A)
A and P(A) are always equal in cardinality
P(A) is always finite
A is uncountable if P(A) is empty
Cantors theorem proves that for any set A, the power set P(A) has strictly greater cardinality than A itself, so no bijection exists. This applies even if A is infinite. It underpins hierarchy of infinities. Details at Cantors theorem.
Which ZF axiom guarantees the existence of a set containing exactly two given sets A and B?
Pairing axiom
Union axiom
Replacement axiom
Infinity axiom
The Pairing Axiom in ZermeloFraenkel set theory asserts that for any sets A and B, there is a set {A, B}. It does not address unions or infinite sets. Replacement is about images under functions. More at ZF axioms.
How is the union over an indexed family {A_i} for i in I defined?
{ x | there exists i in I with x in A_i }
{ x | for all i in I, x in A_i }
?_{i in I} A_i
The set of all I-tuples from each A_i
The union of an indexed family is the set of elements belonging to at least one member set in the family. It uses existential quantification over the index set I. The universal condition defines intersection instead. See Indexed families.
What is Russells paradox about?
The set of all sets that do not contain themselves leads to a contradiction
The cardinality of the reals is greater than that of the naturals
No set can be a member of itself
Every function has an inverse
Russells paradox arises when considering the set of all sets that are not members of themselves, leading to a contradiction if it contains itself. It challenged naive set comprehension. It motivated axiomatic restrictions. More at Russells paradox.
Which property is not required for a relation to be an equivalence relation?
Antisymmetric
Reflexive
Symmetric
Transitive
An equivalence relation must be reflexive, symmetric, and transitive. Antisymmetry is a requirement for partial orders, not equivalence relations. Confusing these properties can lead to mistakes in relation classification. For more, visit Equivalence relation.
In cardinal arithmetic, the continuum is often denoted by which expression?
2^??
??
??
c!
The cardinality of the continuum (the real numbers) is 2^??, where ?? is the cardinality of the naturals. This follows from Cantors diagonal argument. ?? is the next cardinal and not necessarily equal to the continuum. See Continuum.
What does it mean for two sets to be equipotent?
There is a bijection between them
They have equal cardinality if finite only
Their intersection is nonempty
One is a subset of the other
Equipotent sets have the same cardinality, established by the existence of a bijection. This applies to both finite and infinite sets. Intersection or subset relations are unrelated to equipotence. More at Equipotence.
Which of these sets is uncountable?
The real numbers between 0 and 1
The natural numbers
The integers
The rationals
The interval (0,1) is uncountable by Cantors diagonalization argument. Naturals, integers, and rationals are all countable sets. Uncountability means no bijection to the naturals exists. See Uncountable set.
Which axiom schema in ZermeloFraenkel set theory restricts comprehension to avoid forming "the set of all sets"?
Axiom schema of separation
Axiom of comprehension
Axiom of pairing
Axiom of power set
The Axiom Schema of Separation (also called Specification) in ZF limits set formation by allowing subsets of existing sets defined by a property. It replaces naive comprehension to block paradoxical sets like the set of all sets. Pairing and power set are unrelated to this restriction. More at ZF axioms.
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Study Outcomes

  1. Analyze Fundamental Set Theory Concepts -

    Identify and articulate the definitions of sets, subsets, and elements, and explain how these basic components form the foundation of set theory.

  2. Apply Set Operations -

    Perform unions, intersections, differences, and complements to solve problems and demonstrate proficiency with common set operations.

  3. Interpret Venn Diagrams -

    Visually represent and analyze relationships between sets using Venn diagrams, identifying overlaps, disjoint regions, and universal sets.

  4. Evaluate Membership and Subset Relations -

    Determine element inclusion, distinguish between proper and improper subsets, and calculate set cardinalities in various contexts.

  5. Construct Logical Proofs Involving Sets -

    Formulate direct proofs, proofs by contrapositive, and proofs by contradiction to validate set-theoretic statements and logical arguments.

  6. Self-Assess Discrete Math Skills -

    Gauge your proficiency in logic and set theory through scored questions, and identify areas for further practice and improvement.

Cheat Sheet

  1. Set Operations Mastery -

    Understand the union (A ∪ B), intersection (A ∩ B) and set difference (A\B) operations and how they combine sets. For example, A ∪ B includes every element in A or B, while A ∩ B shows only common elements. Use the mnemonic "U sweeps up, ∩ zooms in" to recall which region each symbol covers.

  2. Subset & Power Set Principles -

    Recognize that A is a subset of B (A ⊆ B) if every element of A appears in B, and the power set P(A) contains all subsets of A. If |A|=n, then |P(A)|=2❿ (e.g., A={x,y} gives P(A)={∅,,,{x,y}}). This fact comes straight from standard discrete math texts such as MIT OpenCourseWare.

  3. Venn Diagrams & Inclusion - Exclusion -

    Visualize up to three sets in overlapping circles to solve counting problems by region. Apply the formula |A ∪ B| = |A| + |B| - |A ∩ B| to avoid double-counting common elements. This principle extends to three sets: |A ∪ B ∪ C| = Σ|A| - Σ|A ∩ B| + |A ∩ B ∩ C|.

  4. De Morgan's Laws for Complements -

    Use De Morgan's Laws: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ to simplify complements in proofs. Translating logical statements into set notation (e.g., "not (P or Q)" becomes "Pᶜ ∩ Qᶜ") aids both logic quizzes and set theory quiz problems. These identities are fundamental in courses like Stanford's CS103.

  5. Proof Techniques: Double Inclusion -

    Master the double-inclusion method to prove two sets are equal: first show A ⊆ B by picking an arbitrary x∈A and proving x∈B, then reverse. For instance, to prove A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), chase elements via definitions. Element-wise proofs are a staple in discrete math and logic quiz practice.

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