EXAM 1 & EXAM 2Performance Expectations MAT 231

Items appearing in red are covered on exam 2 but not on exam 1.

LOGIC

Definitions: Be able to define each of the following carefully and precisely.

compound statement
logical connectives using truth tables
logical equivalence
converse, inverse, contrapositive
valid argument

Know how to do

complete a truth table
negate a proposition
prove or disprove a statement with quantifiers
indirect proof

Understanding: Be able to demonstrate a thorough understanding of each of the following.

meaning of "true" and "false" in mathematics
simple statement, logical connectives
basic algebraic properties of conjunction, disjunction, negation
relationship between converse, inverse, contrapositive
premise, conclusion, argument
modus ponens, law of syllogism
propositional function
universal and existential quantifiers

SETS AND FUNCTIONS

Definitions: Be able to define each of the following carefully and precisely.

subset, proper subset, power set
union, intersection, relative complementation, symmetric difference, disjoint
cartesian product, relation, function, domain, range, codomain, single-valued
1-1, onto, 1-1 correspondence, injection, surjection, bijection
inverse relation, composite functions, commutative diagrams
countable and uncountable sets

Know how to do

prove two sets are equal
prove one set is a subset of another
prove a function is 1-1, onto, or a 1-1 correspondence by
• using its equation
• looking at its graph or mapping diagram
• looking at its ordered pairs
determine if an inverse relation is a function
determine if a diagram commutes
show that a set is countably infinite

Understanding: Be able to demonstrate a thorough understanding of each of the following.

notation for sets and special sets (natural numbers, etc.)
specification of sets
well-defined set, well-defined function
index set
singleton set, n-tuple
image, preimage, functional notation (f: X--->Y)
special functions: Dirichlet function, identity, inclusion,
characteristic function, sequence, binary operation, restriction, extension

INDUCTION AND RELATIONS

Definitions and statements: Be able to define each of the following concepts and to state each of the following theorems carefully and precisely.

LWO, PMI, strong PMI
relation, equivalence relation, equivalence class, equivalence class representative, natural map
partition, cell
partition induced by an equivalence relation
equivalence relation induced by a partition
partial ordering, partially ordered set
total ordering, totally ordered set
well-ordering, well-ordered set
linear order, linearly ordered set

Know how to do

PMI, strong PMI, extended versions of PMI and strong PMI
describe equivalence classes of an equivalence relation
prove whether a given relation is an equivalence relation, partial ordering, total ordering, well-ordering, or linear order

Understanding: Be able to demonstrate a thorough understanding of each of the following.

induction hypothesis, induction variable, initial segment
reflexive, anti-symmetric, transitive, comparable, trichotomous