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Question 1
Discrete MathFoundation

Discrete Math practice question 1

Discrete Math practice topics

Question sets mix direct skills with unfamiliar applications so method selection becomes part of the practice.

01

Logic and proof methods

Build converses and contrapositives, and choose between direct proof, contraposition, contradiction, and induction.

02

Sets, relations, and functions

Use inclusion-exclusion on set sizes, and count or classify functions as injective, surjective, or bijective.

03

Counting and recurrences

Choose between permutations and combinations, and unroll a recurrence relation term by term to find a later value.

04

Graphs and number theory

Apply the handshake lemma and connectivity rules, and use the Euclidean algorithm for gcd, lcm, and modular arithmetic.

Discrete Math practice questions with worked answers

These 8 questions are printed in full on this page, with every step of the arithmetic written out. Cover the options, solve the question on paper first, and only then open the worked answer to compare your method with the one shown.

Question 1: What is the contrapositive of 'If it rains, then the game is cancelled'?

Foundation level · logic and proof methods

  1. If the game is not cancelled, then it did not rain
  2. If it does not rain, then the game is not cancelled
  3. If the game is cancelled, then it rained
  4. It rains and the game is not cancelled
Show the worked answer
  1. Write the statement as P implies Q, with P = 'it rains' and Q = 'the game is cancelled'.
  2. The contrapositive is 'not Q implies not P'.
  3. Not Q is 'the game is not cancelled', and not P is 'it did not rain'.
  4. So the contrapositive reads: if the game is not cancelled, then it did not rain. It is always true exactly when the original is true.

Answer: If the game is not cancelled, then it did not rain

Question 2: To prove 'if n^2 is even then n is even' by contraposition, what statement do you prove instead, and does it hold?

Exam level · logic and proof methods

  1. Prove 'if n is odd then n^2 is odd', and it holds
  2. Prove 'if n is even then n^2 is even', and it holds
  3. Prove 'if n^2 is odd then n is even', and it fails
  4. Prove 'n^2 is even and n is odd', and it holds
Show the worked answer
  1. The contrapositive of 'n^2 even implies n even' is 'n odd implies n^2 odd'.
  2. Write an odd number as n = 2k + 1 for some whole number k.
  3. Square it: n^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1.
  4. That is 2 times a whole number plus 1, so n^2 is odd. Example: 7^2 = 49, which is odd.

Answer: Prove 'if n is odd then n^2 is odd', and it holds

Question 3: Set A has 18 elements, set B has 12 elements, and 5 elements are in both. How many elements are in A union B?

Foundation level · sets, relations, and functions

  1. 25
  2. 30
  3. 35
  4. 20
Show the worked answer
  1. Adding 18 + 12 = 30 counts the 5 shared elements twice.
  2. Inclusion-exclusion says |A union B| = |A| + |B| - |A intersect B|.
  3. So the count is 30 - 5.
  4. 30 - 5 = 25 elements in the union.

Answer: 25

Question 4: How many different functions are there from a set of 3 elements to a set of 4 elements?

Exam level · sets, relations, and functions

  1. 64
  2. 12
  3. 81
  4. 24
Show the worked answer
  1. A function must send each input to exactly one output, and the choices are independent.
  2. The first input has 4 possible outputs, and so does the second, and so does the third.
  3. The total is 4 x 4 x 4 = 4^3.
  4. 4^3 = 64 different functions.

Answer: 64

Question 5: How many different 4-person committees can be chosen from 10 people?

Exam level · counting and recurrences

  1. 210
  2. 5040
  3. 40
  4. 120
Show the worked answer
  1. A committee has no order, so this is a combination: 10 choose 4.
  2. 10 choose 4 = (10 x 9 x 8 x 7) / (4 x 3 x 2 x 1).
  3. The top is 10 x 9 x 8 x 7 = 5040, and the bottom is 24.
  4. 5040 / 24 = 210 committees.

Answer: 210

Question 6: A sequence has a1 = 2 and a(n) = 3a(n-1) + 1. What is a4?

Challenge level · counting and recurrences

  1. 67
  2. 22
  3. 40
  4. 81
Show the worked answer
  1. Apply the rule one step at a time, starting from a1 = 2.
  2. a2 = 3(2) + 1 = 7.
  3. a3 = 3(7) + 1 = 22.
  4. a4 = 3(22) + 1 = 66 + 1 = 67.

Answer: 67

Question 7: A graph has 7 vertices and every vertex has degree 4. How many edges does the graph have?

Foundation level · graphs and number theory

  1. 14
  2. 28
  3. 7
  4. 11
Show the worked answer
  1. Add up all the degrees: 7 vertices x degree 4 = 28.
  2. The handshake lemma says the degree total counts every edge twice, once at each end.
  3. So the number of edges is 28 / 2.
  4. 28 / 2 = 14 edges.

Answer: 14

Question 8: Find gcd(84, 126), then use it to find lcm(84, 126).

Challenge level · graphs and number theory

  1. gcd 42 and lcm 252
  2. gcd 21 and lcm 504
  3. gcd 42 and lcm 126
  4. gcd 6 and lcm 1764
Show the worked answer
  1. Use the Euclidean algorithm: 126 = 1 x 84 + 42, then 84 = 2 x 42 + 0.
  2. The last non-zero remainder is the gcd, so gcd(84, 126) = 42.
  3. For any two numbers, gcd x lcm = the product of the numbers: 84 x 126 = 10584.
  4. lcm = 10584 / 42 = 252. Check: 252 = 3 x 84 and 252 = 2 x 126.

Answer: gcd 42 and lcm 252

How the three Discrete Math difficulty levels differ

The difficulty buttons above change what a question asks of you, not just the size of the numbers. Each example below is taken from the question set on this page.

Difficulty levels for Discrete Math practice, with an example question from this page.
LevelWhat it testsExample questionTime target
FoundationOne skill at a time, with the numbers kept small enough to check in your head.What is the contrapositive of 'If it rains, then the game is cancelled'?About 1 minute
ExamThe wording of a real test paper: pick the method first, then carry out two or three steps.To prove 'if n^2 is even then n is even' by contraposition, what statement do you prove instead, and does it hold?2 to 3 minutes
ChallengeTwo ideas combined, or a result you have to interpret after the calculation ends.A sequence has a1 = 2 and a(n) = 3a(n-1) + 1. What is a4?4 to 5 minutes

How to practice Discrete Math effectively

Begin without notes and explain your choice before checking. For every miss, identify whether the cause was a definition, setup, calculation, interpretation, or time decision. Re-solve the question from a blank page, then return to the same skill in a mixed set tomorrow.

What your Discrete Math answer review should show

A useful review shows more than the correct option. Compare the method with your first attempt, locate the earliest incorrect decision, and write one rule that would prevent the same error in a new Discrete Math problem.

Move from mixed Discrete Math questions to a complete course

This page targets flexible Discrete Math question practice. When you need a syllabus-aligned sequence with unit selection, use Discrete Mathematics practice by unit and return here later for mixed retrieval.

Questions about Discrete Math practice

When should I change the difficulty?

Move up after you can solve several questions accurately without hints and explain the method. Move down for one short set when errors show that a definition or setup is still uncertain.

How often should I practice?

Short sessions on several days usually build stronger recall than one long session. Revisit missed Discrete Math skills the next day, then mix them with older topics later in the week.

Where can I review the lessons in order?

Use Discrete Mathematics practice by unit for a syllabus-aligned sequence with unit selection, practice, and a complete answer review.

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