You can work with divisibility, greatest common divisors, primes and clock arithmetic.
Weeks 13-14
Discrete Mathematics course units
Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.
01
Logic
Weeks 1-2
You can build truth tables, negate a statement correctly, and check whether an argument is valid. You can also translate an English sentence into symbols.
Build the full truth table for an if-then statement.
Negate 'all cats are grey' as 'some cat is not grey'.
Apply modus ponens and spot the converse mistake.
Worked example
When is 'if p then q' false?
Check all four rows of the truth table
Only the row with p true and q false fails
AnswerIt is false only when p is true and q is false
Most common mistakeReading 'if p then q' as 'if q then p': from 'if it rains, the ground is wet' concluding 'the ground is wet, so it rained', which any sprinkler disproves.
You can prove a claim directly, by contradiction, by contrapositive, or by induction. You can also knock a claim down with a single counterexample.
Prove odd plus odd is even by writing both numbers in 2k + 1 form.
Run an induction proof with a base case and an inductive step.
Disprove 'every prime is odd' by naming the number 2.
Worked example
Prove that the sum of two odd numbers is even
Let a = 2m + 1 and b = 2n + 1
a + b = 2m + 2n + 2 = 2(m + n + 1)
AnswerIt is 2 times a whole number, so it is even
Most common mistakeStopping after examples: 3 + 5 = 8 and 7 + 9 = 16 show the pattern but prove nothing about the infinitely many pairs you did not test.
You can work with unions, intersections and complements, and test a relation for being reflexive, symmetric or transitive. You can also check whether a relation is a function.
Compute the size of a union with inclusion-exclusion.
Test whether 'is a sibling of' is transitive.
Build the equivalence classes of a relation.
Worked example
Set A has 20 members, set B has 15, and 6 are in both. How big is the union?
Size of union = 20 + 15 - 6
35 - 6
Answer29
Most common mistakeAdding the two sizes and stopping: reporting 35, which counts each of the 6 shared members twice.
You can count arrangements and selections without listing them, using products, permutations and combinations. You can also use the pigeonhole idea to prove something must repeat.
Decide whether order matters before choosing permutations or combinations.
Compute the number of 3-person teams from a group of 10.
Use the pigeonhole principle to show two of 13 people share a birth month.
Worked example
How many 3-person teams can be picked from 10 people?
Order does not matter, so use combinations
(10 x 9 x 8)/(3 x 2 x 1) = 720/6
Answer120
Most common mistakeUsing permutations by mistake: reporting 10 x 9 x 8 = 720, which counts the same three people once for each of their 6 orders.
You can work with divisibility, greatest common divisors, primes and clock arithmetic. You can also solve the simple congruences that coding schemes rely on.
Run the Euclidean algorithm to find a greatest common divisor.
Reduce a large power in modular arithmetic by finding its cycle.
Find the inverse of 3 modulo 7.
Worked example
Find the greatest common divisor of 252 and 105
252 = 2(105) + 42
105 = 2(42) + 21, then 42 = 2(21) + 0
Answer21
Most common mistakeStopping at the first remainder: reporting 42 as the answer, even though 42 does not divide 105 evenly.
Before timing yourself, check whether you can explain Logic from a blank page. Then connect it to Proof methods. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Discrete math calculator can test calculations and representations used in Discrete Mathematics. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from Discrete Mathematics practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Start with Logic if you are following the full sequence. If that unit feels automatic, open the Discrete Mathematics practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Discrete Mathematics test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Discrete math calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
