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Discrete Math Fundamentals Cheat Sheet
A printable reference covering sets, logic, counting rules, permutations, combinations, graphs, and sequences for grades 10-12.
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Discrete math studies objects that are counted, listed, connected, or arranged in separate steps. This cheat sheet helps students organize the core tools used in computer science, data analysis, scheduling, networks, and decision making. It focuses on clear rules for sets, logic, counting, graphs, and sequences. These ideas support problem solving when quantities are whole, finite, or structured rather than continuous. The most important skills are describing sets, applying logical statements, counting outcomes without listing every case, and modeling relationships with graphs. Counting formulas such as n!, nPr, and nCr help solve arrangement and selection problems. Graph vocabulary such as vertex, edge, degree, path, and cycle helps describe networks. Sequences and recurrence rules show how patterns grow from one step to the next.
Key Facts
- The union of two sets includes all elements in either set, written A union B.
- The intersection of two sets includes only elements in both sets, written A intersection B.
- The complement of a set A includes all elements in the universal set that are not in A, written A'.
- The multiplication principle says that if one task has m choices and the next has n choices, then both tasks have m x n total outcomes.
- A permutation counts ordered arrangements, and nPr = n! / (n - r)!.
- A combination counts unordered selections, and nCr = n! / (r! (n - r)!).
- In a finite graph, the sum of all vertex degrees equals 2E, where E is the number of edges.
- An arithmetic sequence has common difference d and formula a_n = a_1 + (n - 1)d.
Vocabulary
- Set
- A set is a well-defined collection of distinct objects called elements.
- Proposition
- A proposition is a statement that is either true or false, but not both.
- Permutation
- A permutation is an arrangement of objects where order matters.
- Combination
- A combination is a selection of objects where order does not matter.
- Graph
- A graph is a structure made of vertices connected by edges.
- Recurrence Relation
- A recurrence relation defines each term of a sequence using one or more earlier terms.
Common Mistakes to Avoid
- Using a permutation when order does not matter is wrong because it counts the same group multiple times in different orders.
- Using a combination when order matters is wrong because it ignores different arrangements that should be counted separately.
- Forgetting the factorial meaning in n! is wrong because n! means n x (n - 1) x (n - 2) x ... x 1, not n squared or n times 10.
- Counting overlapping sets by simple addition is wrong because elements in both sets get counted twice unless the intersection is subtracted.
- Assuming every graph path is a cycle is wrong because a cycle must start and end at the same vertex without repeating other vertices.
Practice Questions
- 1 A password uses 2 letters followed by 3 digits. If letters and digits can repeat, how many passwords are possible?
- 2 How many ways can 4 students be chosen from a group of 12 students for a committee?
- 3 Find the 10th term of the arithmetic sequence 7, 11, 15, 19, ...
- 4 Explain whether arranging 5 books on a shelf should be modeled with a permutation or a combination, and justify your choice.