REVIEW FOR TEST 1, MAT 243, FALL 2013
The exam covers sections 1.1, 1.3, 1.4, 1.5, 1.6, 1.7, 2.1, 2.2, 2.3, and 2.4.
Test 1, October 2, 2013
Think of this review as a starting point for studying for Exam 1.
You will be asked for definitions on the exam. You will be asked to prove statements on
the exam.
General outline
(1) Logic.
• Propositions
• Operations: conjunction, disjunction, biconditional, conditional.
• Different ways to write the conditional.
• Propositional equivalence, tautologies, converse, contrapositive.
• Quantifiers: truth values, negations, positive form, and translations.
• Rules of inference.
(2) Methods of Proof.
• Direct and indirect proofs of implication.
• Proof by contradiction.
• Proof of a biconditional.
• Cases.
• Counterexamples.
(3) Sets.
• Notation, the power set, Cartesian products, size of a set
• Operations on sets: union, intersection, complement.
• Disjoint sets, empty set.
(4) Functions.
• one-to-one and onto
• composition
• inverse
(5) Series and sequences
(a) sequences defined explicitly (closed formula).
(b) P
sequences defined through a recurrence relation
(c)
notation
(d) sum various series
Section by section, minimum preparation
1.1: Know the logical connectives (negation, conjunction, disjunction, exclusive or and implication) and their truth tables. Review all different ways we can express p → q .
Practice translation of English sentences into logical expressions.
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1.3: Know how to determine whether two propositions are equivalent by using truth tables
or other techniques. Learn De Morgan’s laws. The expressions in Table 7 on page 28
are very useful.
1.4: Know the meaning of the quantifiers “for all x” and “there exists an x.” Find truth values of logical statements with quantifiers and be able to explain your answer. Practice
translation of English sentences into logical expressions using logical connectives and
quantifiers. Understand the role of the given domain (universe of discourse) when
translating sentences (see Example 23 on page 48). Understand how negations of
quantified statements are formed. Table 1 and 2 on pages 41 and 47 are very useful.
Understand the difference between statements using “and” and statements using “if
... then.”
1.5: Understand the idea of nested quantifiers. Remember the difference between the statements when you change the order of the quantifiers. Be able to find truth values
of logical statements with nested quantifiers and know how to verify your answer.
Practice translations of English sentences into logical expressions with two and three
variables. Understand how to express statements with “exactly one” and “exactly
two” in it. Understand how negations of nested quantifiers are formed. Table 1 on
page 60 is very useful. Be able to find negation and express it in an English sentence.
1.6: Understand the difference between a True proposition and a valid argument. Know
the main rules of inference and the common fallacies. Know how to translate into
symbolic form and recognize a valid argument or a fallacy.
1.7: Remember the three ways of proving implications (direct and indirect proof and proof
by contradiction). Practice constructing proofs for simple mathematical statements
as in Ex. 1-9 and examples from the homework and class notes and also read about
mistakes in Examples 10-14.
2.1: Know the basic terminology in set theory, subsets, subsets, Cartesian product of sets,
and know how to find the power set of a set (see Examples 14 and 15).
2.2: Operations with sets and the Venn diagrams corresponding to them. Know how to
prove that one set is a subset of the other (by showing that both sides is the subset
of the other or by switching to logic). See Examples 10 through 14. Understand the
generalized union and intersection of a collection of sets (see Examples 16 and 17).
2.3: Know the definitions of one-to-one and onto. Know how to prove that a given function
is or is not one-to-one and/or onto. Know how to compose functions. Know the
definition of an inverse function and how to compute it.
2.4: Be able to compute a few terms of a seqence, which may be defined explicity or through
a recurrence relation. Know the sigma notation and how to sum various series.
A few questions
(1) Let p be the proposition “I will do every exercise in the this book” and q be the
proposition “I will get an ‘A’ in this course.” Express each of these as a combination
of p and q.
(a) I will get an ‘A’ in this course and I will do every exercise in this book.
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(b) Either I will not get an ‘A’ in this course or I will or do every exercise in this
book.
(2) Find the truth table of the compound proposition
(p ∨ q) → (p ∧ ¬r).
(3) Show that these compound propositions are tautologies.
(a) (¬q ∧ (p → q)) → ¬p
(b) ((p ∨ q) ∧ ¬p) → q
(4) Give the converse and the contrapositive of these conditional statements.
(a) If it rains today, then I will drive to work.
(b) If |x| = x, then x ≥ 0.
(c) If n is greater than 3, then n2 is greater than 9.
(5) This is a problem from Rosen.
Suppose that on an island there are three types of people, knights, knaves, and
normals (also known as spies). Knights always tell the truth, knaves always lie, and
normals sometimes lie and sometimes tell the truth. Detectives questioned three
inhabitants of the island - Amy, Brenda, and Claire - as part of the investigation
of a crime. The detectives knew that one of the three committed the crime, but
not which one. They also knew that the criminalwas a knight, and that the other
two were not. Additionally, the detectives recorded these statements: Amy: “I am
innocent.” Brenda: “What Amy says is true.” Claire: “Brenda is not a normal.”
After analyzing their information, the detectives positively identified the guilty party.
Who was it?
(6) Let P (m, n) be the statement “m divides n,” where the domain for both variables
consists of all positive integers. (By “m divides n” we mean that n = km for some
integer k.) Determine the truth values of each of these statements.
(a) P (4, 5)
(b) P (2, 4)
(c) ∀m∀nP (m, n)
(d) ∃m∀nP (m, n)
(e) ∃n∀mP (m, n)
(f) ∀nP (1, n)
(7) Find the negations of these statements.
(a) If it snows today, then I will go skiing tomorrow.
(b) Every person in this class understands mathematical induction.
(c) Some students in this class do not like discrete mathematics.
(d) In every mathematics class there is some student who falls asleep during lectures.
(8) Prove that if x3 is irrational, then x is irrational.
(9) Let A be the set of English words that contain the letter x, and let B be the set of
English words that contain the letter q. Express each of these sets as a combination
of A and B.
(a) The set of English words that do not contain the letter x.
(b) The set of English words that contain both an x and a q.
(c) The set of English words that contain an x but not a q.
(d) The set of English words that do not contain either an x or a q.
(e) The set of English words that contain an x or a q, but not both.
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(10) Show that if A is a subset of B, then the power set of A is a subset of the power set
of B.
(11) Suppose that A and B are sets such that the power set of A is a subset of the power
set of B. Does it follow that A is a subset of B?
(12) Show that if A and B are sets, then A − (A − B) = A ∩ B.
(13) Let A,B, and C be sets. Show that (A−B)−C is not necessarily equal to A−(B−C).
(14) Suppose that A,B,C, and D are sets. Prove or disprove that (A − B) − (C − D) =
(A − C) − (B − D).
(15) Show that if A and B are finite sets, then |A ∩ B| ≤ |A ∪ B|. Determine when this
relationship is an equality.
(16) Let A and B be sets in a finite universal set U . List the following in order of increasing
size.
(a) |A|, |A ∪ B|, |A ∩ B|, |U |, |∅|
(b) |A − B|, |A| + |B|, |A ∪ B|, |∅|
(17) Evaluate
200
X
1
( )k
3
k=0
Leave your answer as a fraction, do not simplify and do not evaluate exponents.
(18) For each statement, circle TRUE or FALSE.
(a) Let f : Z → Z be defined by f (n) = n3 .
(i) f is one-to-one. TRUE or FALSE.
(ii) f is onto. TRUE or FALSE.
(b) Let f : Z → Z be defined by f (n) = d n3 e.
(i) f is one-to-one. TRUE or FALSE.
(ii) f is onto. TRUE or FALSE.
(19) Prove ¬p, assuming the following:
(a) ¬(¬q → ¬r)
(b) ¬(¬q → ¬r) → ¬q
(c) ¬(p ∧ (¬p → ¬s))
(d) s → (¬p ∨ q)
(e) ¬(p ∧ (¬p → ¬s)) → s
You must use the rules of inference to justify your work.
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