Look at the objectives for Chapter: 1, 2, 3, 4, 5, 6, 7, 10, 11
1.1.1 Given a statement form made
up of statement variables and logical connectives (such as not, and, or)
construct
a truth table for the given statement form. (Exercise Set 1.1:
12-16)
1.1.2 Given two statement forms use truth tables to
determine whether or not the statement forms are logically
equivalent. (Exercise Set 1.1: 17-26)
1.1.3 Apply De Morgan's Laws to write negations of compound statements. (Exercise Set 1.1: 27-36)
1.1.3 Use logical equivalences to simplify statement forms. (Exercise Set 1.1: 41-47)
1.1.4 Translate English statements and arguments written in English into symbolic form. (Exercise Set 1.1: 1-4, 6-9)
1.2.1 Verify logical equivalences involving conditional statements. (Exercise Set 1.2: 14, 29-32)
1.2.2 Write negations of conditional statements. (Exercise Set 1.2: 15-16)
1.2.3 Correctly express conditional statements a variety of ways. (Exercise Set 1.2: 1-4, 25-28, 36-37)
1.2.4 Given a conditional statement, state its converse, contrapositive, and inverse.
1.2.5 Correctly interpret necessary and sufficient
conditions, and convert those conditions to "if __ then" form.
(Exercise Set 1.2: 34-35, 38-39)
1.3.1 Test argument forms for validity using truth tables. (Exercise Set 1.3: 6-10)
1.3.2 Recognize and apply valid argument forms. (Exercise Set 1.3: 1-5, 23-31)
1.3.3 Recognize invalid argument forms. (Exercise Set 1.3: 23-31)
1.3.4 Specify the following rules of inference both
in symbols and in English: modus ponens, modus tollens,
disjunctive syllogism, hypothetical syllogism, and rule of contradiction.
1.3.5 Given a set of premises and a conclusion, use
valid argument forms to deduce the conclusion from
the premises and give a reason for each step. (Exercise Set 1.: 49-43)
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2.1.1 Determine the truth values of universal and existential statements. (Exercise Set 2.1: 1-7, 28)
2.1.2 Translate universal and existential statements
from formal to informal language and also from informal
to formal language. (Exercise Set 2.1: 9-13)
2.1.3 Recognize and apply equivalent forms of universal and existential statements. (Exercises 2.1: 14-17)
2.1.4 Write negations of quantified statements. (Exercise Set 2.1: 19-27, 29-36)
2.2.1 Translate multiply quantified statements from
formal to informal language and from informal to
formal language. (Exercise Set 2.2: 3-8, 11-17)
2.2.2 Determine truth values of multiply quantified statements. (Exercise Set 2.2: 18-21)
2.2.3 Write negations of multiply quantified statements. (Exercise Set 2.2: 3-8, 11-17)
2.2.4 Write the contrapositive, converse, and inverse
of universal conditional statements.
(Exercise Set 2.2: 22-29)
2.2.5 Rewrite universal conditional statements using the words necessary, sufficient, and only if.
2.2.6 Translate among the variants of universal quantified statements. (Exercise Set 2.2: 31-34, 35-38)
2.3.1 State the rule of universal instantiation.
2.3.2 Draw conclusions using universal modus ponens (modus tollens). (Exercise Set 2.3: 1-19)
2.3.3 Use universal modus ponens (modus tollens) in proofs. (Exercise Set 2.3: 27-28)
2.3.4 Use diagrams to test arguments for validity (invalidity). (Exercise Set 2.3: 20-26)
2.3.5 Recognize converse errors and inverse errors. (Exercise Set 2.3: 7-19)
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3.1.1 Prove existential statements
using constructive proofs of existence and nonconstructive proof of
existence. (Exercise Set 3.1: 3-7)
3.1.2 Prove universal statements using the method
of generalizing from the generic particular or
the method of direct proof. (Exercise Set 3.1: 11-14,
25-41)
3.1.3 Disprove universal statements by counterexample. (Exercise Set 3.1: 15, 16, 25-41)
3.2.1 Prove or disprove quantified statements about rational numbers. (Exercise Set 3.2: 13-22)
3.3.1 Prove or disprove quantified statements about divisibility properties. (Exercise Set 3.3: 14-26, 41)
3.4.1 Write proofs using the technique of proof by cases. (Exercise Set 3.4: 18, 20-26, 36)
3.6.1 Write proofs using the technique of proof by contradiction. (Exercise Set 3.6: 6, 10, 17-21)
3.6.2 Write proofs using the technique of proof by contraposition. (Exercise Set 3.6: 7, 8, 10)
3.7.1 Prove that the square root of 3 is irrational.
3.7.1 Prove that there exists an infinite number of primes.
3.7.2 Prove or disprove statements like the following: Exercise Set 3.7: 11, 12, 21, 23
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4.1.1 Demonstrate knowledge of the terminology
and symbolism we use in the study of sequences by
(a) Finding
terms of sequences defined by explicit formulas,
(b) Finding
an explicit formula to git given initial terms,
(c) Interpreting,
using, and manipulating summation notation,
(d) Interrpreting,
using, and manipulating product notation,
(e) Using
properties of summation and product, and
(f) Computing
with factorials.
(Exercise Set
4.1: 1-7 odd, 8-12 even, 18-26 even, 33-41 odd, 45, 48-56 even)
4.2.1 State the Principle of Mathematical Induction.
4.2.2 Prove properties of selected arithmetic and geometric series
by the technique of mathematical induction.
(Exercise Set
4.2: 3, 5, 6, 8, 9, 10, 12, 15, 27)
4.3.1 Prove divisibility properties using the technique of mathematical induction. (Exercise Set 4.3: 8, 10)
4.3.2 Establish the true of properties on inequality using the technique of mathematical induction. (Exercise Set 4.3: 16, 17)
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5.1.1 Demonstrate knowledge of the basic symbols
and definitions of set theory presented in Section 5.1.
(pp. 231-239
only) (Exercise Set 5.1: 1-18 all)
5.2.1 Demonstrate knowledge of, and ability to apply, properties of sets. (Exercise Set 5.2: 30-36)
5.2.1 Prove that a specified set is a subset of another specified set.
5.2.2 Prove that a pair of specified sets are equal.
5.2.3 Prove set identities. (e.g. Those in Theorem 5.2.2)
5.2.3 Establish the truth or falsity of statements about sets. (Exercise Set 5.2: 9-24)
5.3.1 Demonstrate knowledge of, and the ability to apply, the
notation, terminology and concepts
of Section 5.3.
(Exercvise Set 5.3: 1, 35, 38, 40, 44, 46)
5.3.2 Derive set identities from known properties. (Exercise Set 5.3: 21-28)
5.3.3 Prove or disprove statements about sets or power sets. (Exercise Set 5.3: 7-17)
5.3.4 Define: Boolean Algebra
5.3.5 Prove properties of Boolean algebras. (Exercise Set 5.3: 48)
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6.2.1 Apply the multiplication
rule to determine the number of ways a sequence of events may
occur. (Exercise Set 6.2: 1-10, 31)
6.2.2 Use properties of permutations to solve problems
related to arragements.
(Exercise Set 6.2: 33, 35)
6.2.3 Apply the addition rule, difference rule, or
exclusion/inclusion rule in determining the number
of elements in specified sets. (Exercise Set 6.2: 23, 24)
6.3.1 Use counting techniques to solve counting problems such as 6,7,8 in Exercise Set 6.3:
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7.1.1 Define each of the following:
function f from a set X to a set Y, domain
of f, co-domain of f,
value
of f at x, image of f at x, range of f,
image of X under f, inverse image of y.
7.1.2 Define what we mean when we say two functions are equal.
7.1.3 Show how to determine whether or not a rule or formula is
well-defined and specifies a function.
(Exercise Set
7.1: 28, 29)
7.1.4 Demonstrate knowledge, understanding, and ability to apply
the concepts defined in 7.1.1
and 7.1.2.
(Exercise Set 7.1: 1, 3, 6)
7.3.1 Define what it means to say a function is one-to-one or injective.
7.3.2 Define what it means to say a function is onto or surjective.
7.3.3 Demonsatrate knowledge and understanding of the concepts
of one-to-one and onto functions.
(Exercise Set
7.3: 1, 2, 3, 4, 5, 6)
7.3.4 Given a rule for a function determine whether or not the
function is one-to-one and whether
not it is onto
and justify your answer by proving it is (is not) one-to-one (onto).
(Exercise Set
7.3: 8, 10, 11, 12, 13)
7.3.5 Given a one-to-one function, find its inverse. (Exercise Set 7.3: 33, 34, 35)
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10.1.1 Define: Relation R from A to B
10.1.2 Sketch an arrrow diagram for a relation. (Exercise Set 10.1: 23-27)
10.2.1 Define what it means to say a relation is (a) reflexive, (b) symmetric, (c) transitive
10.2.2 Given a relation R, determine whether
or not it is (a) reflexive, (b) symmetric, (c) transitive
(Excercise Set 10.2: 1-8, 12-20)
10.2.3 Demonstrate knowledge of, and ability to apply, properties of congruence modulo n relations.
10.3.1 Define: Equivalence relation
10.3.2 Define: Equivalence class
10.3.3 Given a relation, determine whether or not it is an equivalence relation.
10.3.3 Given an equivalence relation, specify its equivalence classes. (Exercise Set 10.3: 2-7)
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11.1.1 Demonstrate knowledge
and understanding of the basic definitions, terminology, notation
and properties of graph theory. (graph, vertices, edges, endpoints,
edge-endpoint function,
loops, parallel edges, adjacent vertices/edges, isolated vertex, empty
graph, driected
graph, digraph, simple graph, complete graph on n vertices,, complete bipartite
graph
on (m,n) vertices, Km,n, subgraph, degree of a vertex, deg(v),
total degree of a graph.
(Exercise Set 11.1: 1-7 odd, 15, 16, 19, 22, 25a, 28, 33)
11.2.1 Demonstrate knowledge, understanding, and the ability
to apply the basic definitions and
concepts related to paths and circuits of graphs. (walk fron v
to w, path from v to w,
simple path from v to w, closed walk, simple circuit, connected
vertices in a graph,
connected graph, connected component of a graph, Euler circuits, Euler
path from v to w,
Hamiltonial
circuit) (Exercise Set 11.2: 1, 5, 8, 10)
11.2.2 Given a graph determine whether or not it has a Euler curcuit
and justify your answer by
citing a theorem and, in case such a circuit exists, describe the curcuit.
(Exercise Set 11.2: 12-17)
11.2.3 Given a tracing puzzle, formulate the proble in graph theory
terms and solve the puzzle.
(Exercise Set 11.2: 18, 22)
11.3.1 Given a digraph, display its adjacency matrix. Also
given an adjacency matrix for a digraph,
draw a digraph having that adjaceny matrix. (Exercise Set 11.3:
2-5 all)
11.3.2 Show how to use the adjacency matrix for a graph to determine
the number of walks of
length k from one specified vertex to another specified vertex. (Exercise
Set 11.3: 20)