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
- tautology, contradiction
- 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

Exams 1 and 2 - Performance ExpectationsFriske Personal Page

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