MAT 1348 Review Sheet
Some of the Most Important Concepts, Methods, Theorems, and Questions. For a list
of all the topics covered, see the course webpage.
Note: You must be able to give precise definitions of important concepts, as well as
state important theorems.
• Logic and Proofs
Concepts
1. propositions, logical connectives, equivalence
2. tautology, contradiction, contingency
3. disjunctive normal form
4. rules of inference, arguments; validity of an argument
5. counter-examples
6. types of proof: direct, indirect, by contradiction, proof of equivalence, proof
by cases
Methods
1. truth tables
2. equivalences
3. truth trees
4. proofs
Questions
1. Translate English sentences into compound propositions.
2. Is a given set of propositions consistent?
3. Is a given proposition a tautology/contradiction?
4. Are given propositions equivalent?
5. Find a disjunctive normal form of a given proposition.
6. Is a given argument valid?
7. Prove a given theorem using a specified method of proof.
• Sets
Concepts
1. set, cardinality, subset, equality of sets
2. important number sets: N, Z, Q, R, Z+ , Z− , Q+ , Q− , etc.
3. set operations: union, intersection, complement, difference, symmetric difference
4. power set, cartesian product of sets
5. Venn diagram
Questions
1. Find the cardinality of a set, power set, cartesian product etc.
2. Prove a set identity (using a rigorous proof, the set identities, membership
tables, or Venn diagrams).
3. Prove a theorem involving sets using an appropriate type of proof.
• Functions
Concepts
1. function
2. injective (one-to-one) and surjective (onto) function; bijection
3. invertible function, inverse
4. composition of functions
Questions
1. Determine whether a given function is injective, surjective, invertible, or a
bijection. Prove your answer is correct.
2. Find the inverse of an invertible function.
3. Find the composition of two functions (if it exists).
• Relations
Concepts
1. binary relation from A to B; binary relation on a set A
2. reflexive, symmetric, antisymmetric, transitive relations
3. equivalence relations, equivalence classes, partitions
4. congruences and modular arithmetic
Questions
1. Determine whether a given relation is reflexive, symmetric, antisymmetric, or
transitive.
2. Give an example of a relation with prescribed properties.
3. Prove that a given relation is an equivalence relation.
4. Determine the equivalence classes of a given equivalence relation (or the corresponding partition of the underlying set).
5. Is a given collection of subsets of A a partition of A?
6. Give the equivalence relation corresponding to a given partition.
• Counting
Concepts
1. Product Rule, Sum Rule, Principle of Inclusion – Exclusion, Complementation
Rule
2. Pigeonhole Principle, Generalized Pigeonhole Principle
3. combinations, permutations
4. symbols P (n, r), n!, C(n, r),
n
r
5. binomial coefficients, Binomial Theorem
6. properties of binomial coefficients
Questions
1. How many...? Questions involving one or more of the rules, permutations, and
combinations. Counting objects such as: strings, binary strings, functions,
injective functions, subsets (selections, combinations), arrangements (permutations),...
2. What is the smallest number of ... to guarantee...?
3. Show that in any set of... we can find... such that.... (Pigeonhole Principle)
4. Compute (for small n, r, without calculator): n!, P (n, r), C(n, r), nr
5. The Binomial Theorem. Find the coefficient of ... in the expansion of...
• Mathematical Induction
Concepts
1. Principle of Mathematical Induction
2. Basis of Induction, Induction Step, Induction Hypothesis
Questions
1. Prove ... is true for all integers n ≥... using Mathematical Induction.
2. State the proposition P (n) to be proved.
3. State what is to be proved in the Basis of Induction and the Induction Step.
4. State the Induction Hypothesis (I.H.).
5. Prove the Basis of Induction and Induction Step (and show where I.H. is used
in your proof).
6. Conclude: for all integers n ≥ . . . .P (n), by Mathematical Induction.
• Graphs
Concepts
1. graph, simple graph, directed graph
2. complete graphs, cycles, paths, complete bipartite graphs
3. bipartite graphs
4. subgraphs, isomorphism
5. connected graphs
6. trees, forests
Theorems
1. Handshaking Theorem
2. characterization of bipartite graphs
3. number of edges in a tree
Questions
1. How many edges in a graph with a given degree sequence?
2. Does there exist a graph/simple graph with the given degree sequence?
3. Is a given graph bipartite? (If so, give a proper 2-vertex colouring. If not, find
an odd cycle.)
4. Are given graphs isomorphic? (If so, give an isomorphism. If not, find an
invariant in which they disagree.) Checking a candidate for isomorphism using
adjacency matrices.
5. Is the given graph a tree? A forest?
6. How many edges in a tree with a given number of vertices?
7. Draw a graph with required properties.