Discrete Mathematics Beginner Quiz
30 questions on Discrete Math.
Question 1: What is Discrete Mathematics?
- A. The study of continuous math like Calculus
- B. The study of mathematical structures that are fundamentally discrete rather than continuous, dealing with distinct and separated values β (correct answer)
- C. The study of shapes and geometry
- D. The study of imaginary numbers
Explanation: Unlike calculus which deals with smooth curves, discrete math deals with integers, graphs, and logical statementsβconcepts that can be counted individually.
Question 2: What is a "Set" in Discrete Mathematics?
- A. An ordered list of numbers
- B. An unordered collection of distinct objects, considered as an object in its own right β (correct answer)
- C. A type of graph
- D. A boolean variable
Explanation: A set is a well-defined collection of distinct elements, such as {1, 2, 3}. Duplicates are ignored, and order does not matter.
Question 3: What does the "Intersection" of two sets A and B (A β© B) represent?
- A. All elements in both A and B combined
- B. Only the elements that are present in BOTH set A and set B β (correct answer)
- C. Elements in A but not in B
- D. An empty set
Explanation: The intersection contains only the common elements shared by both sets.
Question 4: What does the "Union" of two sets A and B (A βͺ B) represent?
- A. Only the elements present in both sets
- B. A set containing all elements from A and all elements from B, with duplicates removed β (correct answer)
- C. Elements in B but not in A
- D. A boolean true/false value
Explanation: The union combines all unique elements from both sets into a single new set.
Question 5: What is the "Power Set" of a set S?
- A. The set of all integers
- B. The set containing all possible subsets of S, including the empty set and S itself β (correct answer)
- C. The largest element in the set
- D. A set multiplied by itself
Explanation: If a set has n elements, its Power Set contains 2^n elements. For {1, 2}, the power set is {{}, {1}, {2}, {1, 2}}.
Question 6: In propositional logic, what does the operator AND (Conjunction, β§) require to evaluate to True?
- A. Both propositions must be False
- B. At least one proposition must be True
- C. Both propositions must be True β (correct answer)
- D. Only the first proposition must be True
Explanation: The AND operator (p β§ q) is true if and only if both p and q are true.
Question 7: In propositional logic, what does the operator OR (Disjunction, β¨) require to evaluate to True?
- A. Both propositions must be True
- B. At least one of the propositions must be True β (correct answer)
- C. Both propositions must be False
- D. Exactly one proposition must be True
Explanation: The inclusive OR operator (p β¨ q) is true if p is true, q is true, or both are true.
Question 8: What is an "Implication" (p β q) in logic?
- A. A statement that is False ONLY when p is True and q is False β (correct answer)
- B. A statement that requires both to be True
- C. An AND operator
- D. A set intersection
Explanation: "If p, then q". If you make a promise (p is True) and break it (q is False), the implication is False. Otherwise, it is logically True.
Question 9: What is a "Tautology" in logic?
- A. A statement that is always False
- B. A statement that is always True, regardless of the truth values of its variables β (correct answer)
- C. A recursive function
- D. A graph with no edges
Explanation: An example of a tautology is p β¨ Β¬p ("It is raining OR it is not raining"). It covers all possibilities and is undeniably true.
Question 10: What is De Morgan's Law regarding the negation of an AND statement: Β¬(p β§ q)?
- A.
Β¬p β§ Β¬q
- B.
Β¬p β¨ Β¬q β (correct answer)
- C.
p β¨ q
- D.
p β§ q
Explanation: De Morgan's Laws state that the negation of an AND is the OR of the negations, and vice versa. "Not (A and B)" equals "Not A OR Not B".
Question 11: In combinations and permutations, what does "Permutation" emphasize that "Combination" does not?
- A. The size of the set
- B. The specific ORDER of the chosen elements β (correct answer)
- C. The colors of the objects
- D. The intersection of sets
Explanation: In permutations, order matters (ABC is different from CAB). In combinations, order does not matter (A hand of cards is the same regardless of how you hold them).
Question 12: What is the mathematical formula for n! (n factorial)?
- A.
n + (n-1) + (n-2) ... + 1
- B.
n * n
- C.
n * (n-1) * (n-2) * ... * 1 β (correct answer)
- D.
n / 2
Explanation: Factorials are used extensively in counting and probability to determine the total number of ways to arrange n distinct items.
Question 13: What is a "Function" in discrete math?
- A. A relation where every element in the domain maps to exactly one element in the codomain β (correct answer)
- B. Any random mapping of numbers
- C. A set of edges
- D. A boolean algebra operator
Explanation: For a relation to be a true mathematical function, an input (x) must produce one and only one predictable output (y).
Question 14: What does it mean for a function to be "Injective" (One-to-One)?
- A. All inputs map to the same output
- B. Every element in the codomain is mapped to by at least one element
- C. No two different elements in the domain map to the same element in the codomain β (correct answer)
- D. The domain and codomain are empty
Explanation: In a one-to-one function, every input has a unique output. f(a) = f(b) implies a = b.
Question 15: What does it mean for a function to be "Surjective" (Onto)?
- A. Every element in the domain has an output
- B. Every element in the codomain (target set) is mapped to by at least one element in the domain β (correct answer)
- C. The function is continuous
- D. The function uses geometry
Explanation: In a surjective function, there are no "leftover" or unmapped elements in the target output set.
Question 16: If a set A has 5 elements, how many elements does its Power Set have?
- A. 5
- B. 10
- C. 25
- D. 32 β (correct answer)
Explanation: The formula for the size of a power set is 2^n. For n=5, 2^5 = 32.
Question 17: What does the Pigeonhole Principle state?
- A. Birds must stay in cages
- B. If
n items are put into m containers, with n > m, then at least one container must contain more than one item β (correct answer)
- C. All sets must intersect
- D. Functions must be bijective
Explanation: This simple logic concept proves facts like: "In any group of 367 people, at least two must share the exact same birthday."
Question 18: Which logical operator acts as an "Exclusive OR" (XOR)?
- A. True if both are True
- B. True if one is True, and the other is False β (correct answer)
- C. True if both are False
- D. True always
Explanation: XOR is strict. "You can have soup OR salad" means you get one or the other, but not both, and not neither.
Question 19: What is a "Graph" in discrete mathematics?
- A. A visual chart with X and Y axes representing data trends
- B. A collection of vertices (nodes) connected by edges (links) β (correct answer)
- C. A geometric polygon
- D. A truth table
Explanation: Graph theory in discrete math does not refer to bar charts. It refers to network structures representing pairwise relations between objects.
Question 20: What is the "Degree" of a vertex in an undirected graph?
- A. The temperature of the node
- B. The total number of edges connected to that vertex β (correct answer)
- C. The shortest path to the root
- D. The weight of the node
Explanation: If node A is connected to node B and node C, the degree of node A is 2.
Question 21: What is a "Bipartite Graph"?
- A. A graph with exactly two edges
- B. A graph whose vertices can be divided into two disjoint sets such that every edge connects a vertex in one set to a vertex in the other set β (correct answer)
- C. A graph that loops back on itself
- D. A graph with two root nodes
Explanation: Bipartite graphs are used to model matching problems (e.g., matching a set of Job Applicants to a set of Open Positions).
Question 22: In Boolean Algebra, what is the result of A β§ (A β¨ B)? (Absorption Law)
- A. B
- B. A β (correct answer)
- C. True
- D. False
Explanation: This is the Absorption Law. If A is true, the whole statement is true. If A is false, the whole statement is false. Thus, it simplifies entirely to A.
Question 23: What is the "Cardinality" of a set?
- A. The sum of its numbers
- B. The number of unique elements it contains β (correct answer)
- C. The largest number in the set
- D. Its intersection with another set
Explanation: Cardinality simply means the size of the set. The cardinality of {apples, oranges, bananas} is 3.
Question 24: What is a "Bijective" function?
- A. A function that is neither injective nor surjective
- B. A function that is BOTH injective (one-to-one) and surjective (onto) β (correct answer)
- C. A function that returns boolean values
- D. A graph theory path
Explanation: Bijective functions have a perfect 1-to-1 pairing between elements in the domain and codomain. Because of this, bijective functions are perfectly reversible (invertible).
Question 25: How many permutations are there for arranging the letters A, B, and C?
- A. 3
- B. 6 β (correct answer)
- C. 9
- D. 27
Explanation: Permutations of 3 distinct items is calculated as 3! (3 * 2 * 1) = 6. (ABC, ACB, BAC, BCA, CAB, CBA).
Question 26: What is a "Tree" in the context of graph theory?
- A. A graph with green nodes
- B. A connected, undirected graph with no cycles β (correct answer)
- C. A directed graph with cycles
- D. A set of disconnected nodes
Explanation: Any connected graph without loops/cycles mathematically forms a tree structure. It has exactly V - 1 edges (where V is the number of vertices).
Question 27: What does the notation n C r (or combinations C(n, r)) calculate?
- A. The number of ways to arrange
r items from n items where order MATTERS
- B. The number of ways to choose
r items from n items where order DOES NOT matter β (correct answer)
- C. The factorial of n and r
- D. The power set of n
Explanation: Combinations represent selecting a team of people from a larger group. The formula divides out the redundant permutations because the order of the team members doesn't matter.
Question 28: In probability, what is the probability of rolling a sum of 7 with two standard six-sided dice?
- A. 1/6 β (correct answer)
- B. 1/36
- C. 7/36
- D. 1/12
Explanation: There are 36 total possible outcomes (6 * 6). There are 6 ways to roll a 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). Therefore, 6/36 simplifies to 1/6.
Question 29: What is "Mathematical Induction"?
- A. A physical hardware component
- B. A method of mathematical proof typically used to establish a given statement for all natural numbers β (correct answer)
- C. A way to calculate power sets
- D. A sorting algorithm
Explanation: Induction proves a base case (e.g., n=1), then proves that if it holds for n=k, it must hold for n=k+1, causing a domino effect proving it for all positive integers.
Question 30: What does an "Eulerian Circuit" represent in a graph?
- A. A path that visits every vertex exactly once
- B. A path that starts and ends on the same vertex, and traverses every EDGE exactly once β (correct answer)
- C. A tree traversal method
- D. A matrix calculation
Explanation: The famous Seven Bridges of KΓΆnigsberg problem led Euler to define this. A graph only has an Eulerian circuit if every vertex has an even degree.