Navigation
- Chapter 1: Section 1.1, Section 1.2, Section 1.3, Section 1.4, Section 1.5.
- Chapter 2: Section 2.1, Section 1.2, Section 2.3.
- Chapter 3: Section 3.1, Section 3.2, Section 3.3, Section 3.4, Section 3.5, Section 3.6.
- Chapter 4: Section 4.1, Section 4.2, Section 4.3, Section 4.4, Section 4.6.
- Chapter 5: Section 5.1, Section 5.2, Section 5.3.
- Chapter 6: Section 6.1, Section 6.2, Section 6.3, Section 6.4.
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Section 1.1
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- be able to determine if an argument is valid or not (Example 1.1.1, Problem 1.1.7);
- know the symbols and use of "and", "or" and "not";
- be able to translate statements from English to logical symbols (Example 1.1.2, Problem 1.1.1) and vice-versa (Example 1.1.3, Problem 1.1.6);
- be able to determine if a logical expression is well-formed (Problem 1.1.4).
Problems
Turn in: 3(d), 6(b), 7(b).
Extra Problems: 1, 3(a-c), 6(a), (c), 7(a), (c-d).
Section 1.2
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- know how to write truth tables (Examples 1.2.1, 1.2.2, Problem 1.2.2);
- understand the concept of logical equivalency;
- simplify and prove equivalency using Boolean algebra (Examples 1.2.5, 1.2.6, Problem 1.2.12).
Problems
Turn in: 2(b), 12(b).
Extra Problems: 2(a), 12(a), (c).
Section 1.3
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- be familiar with the notions of sets and its basic operations (union, intersection and subtraction);
- know what the Truth Set of a statement (which depends on a variable) is (Example 1.3.5, Problem 1.3.8).
Problems
Turn in: None.
Extra Problems:2, 4, 6, 8.
Section 1.4
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- know how to use Venn Diagrams (Problems 1.4.6 and 1.4.11);
- prove equality of two sets with logic (Problem 1.4.7).
Problems
Turn in: 6(a), 7(a), 9.
Extra Problems: 2, 6(b), 7(b).
Section 1.5
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- know the truth table of if/then operation (Problem 1.5.4);
- know that $P \rightarrow Q$ is equivalent to $\neg P \vee R$;
- know the contrapositive and converse of if/then statements;
- translate statements with if/then from common speech to logical symbols (Examples 1.5.1, 1.5.3, Problem 1.5.3) and vice-versa;
- simplify and establish equivalency of statements involving if/then (Example 1.5.2, Problems 1.5.5, 1.5.9).
Problems
Turn in: 5, 9.
Extra Problems: 3, 4.
Section 2.1
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand and know how to use quantifiers, including multiple nested quantifiers (Examples 2.1.1, 2.1.4);
- translate statements with quantifiers from common speech to logical symbols (Examples 2.1.2, 2.1.3, Problem 2.1.3) and vice-versa (Problem 2.1.5).
Problems
Turn in: 6.
Extra Problems: 3, 5.
Section 2.2
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- know how to restate negation of quantified statements in positive form (Problem 2.2.2);
- understand and know how to used bounded quantifiers and how to negate them;
- know the ``distributive law'' for quantifiers (pgs. 70 and 71).
Problems
Turn in: 2(b-c), 7.
Extra Problems: 2(a), (d), 5, 10.
Section 2.3
Videos
- Problem 2.3.1(a-c). NOTE: There is a mistake in part (c)! There is a $\in$ symbol that makes no sense. Please read the comments in the video and turn on annotations (click on the gear icon below the video window and click "On" for "Annotations") to see exactly where.
- Problem 2.3.1(d).
- Problem 2.3.10 and 2.3.11. NOTE: There is a mistake in Problem 10! On the second board I copy and $\land$ and an $\lor$, and the error carries over, resulting in an $\cup$ where there should be an $\cap$. Please read the comments in the video and turn on annotations (click on the gear icon below the video window and click "On" for "Annotations") to see exactly where.
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand families of sets (i.e., sets whose elements are sets), power sets and indexed families of sets;
- understand and know how to translate to logic statements unions and intersections of families as above (most problems from 2.3).
Problems
Turn in: 2(c), 12(a-b).
Extra Problems: 2(a-b), (d), 5, 6, 9, 12(c). (Also, take a look at the statements of 14 and 15.)
Section 3.1
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand what a proof is;
- be able to write (Problems 3.1.6 and 3.1.10) and read (Problems 3.1.2) simple proofs;
- be able to prove simple statements of the form $P \rightarrow Q$ (Example 3.1.2, Problems 3.2.2, 3.2.4);
- be able to prove simple statements of the form $P \rightarrow Q$ using the contrapositive (Example 3.1.3, Problems 3.1.10).
Problems
Turn in: 6, 10.
Extra Problems: 2, 15, 16.
Section 3.2
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- be able to prove simple negative statements (i.e., of the form $\neg P$ -- Example 3.2.1, Problem 3.2.7);
- be able to prove statements by contradictions (Examples 3.2.2, 3.2.3)
- be able to use assumptions of the forms $P \rightarrow Q$ and $\neg P$ in proofs (Example 3.2.5, Problem 3.2.9).
Problems
Turn in: 4, 7.
Extra Problems: 2, 9, 12.
Section 3.3
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- be able to prove simple statements involving quantifiers (most problems of 3.3);
- be able to use in proofs assumptions involving quantifiers (most problems of 3.3);
- know the notation ($a \mid b$) and definition for "$a$ divides $b$" and how to prove statements involving divisibility (Theorem 3.3.6, Problem 3.3.18);
Problems
Turn in: 10, 15.
Extra Problems: 2, 4, 18, 21.
Section 3.4
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- be able to prove and use statements connected by "and"s (Example 3.4.1, Problem 3.4.3);
- know how to approach proofs on containment (Problems 3.4.4, 3.4.8, etc.) equalities of two sets (Example 3.4.4, Problem 3.4.16);
- know how to approach proofs on iff statements (Examples 3.4.2, 3.4.3, Theorem 3.4.6, Problem 3.4.8).
Problems
Turn in: 10, 16.
Extra Problems: 3, 8, 24.
Section 3.5
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- know how to prove a statement with an "or" in the conclusion (Examples 3.5.2, 3.5.3, Problem 3.5.8);
- know how to prove a statement with an "or" in the premise (Example 3.5.1, Problem 3.5.17).
Problems
Turn in: 8, 21.
Extra Problems: 9, 13, 17.
Section 3.6
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand the "there exists a unique" quantifier, including how to write it without using the $\exists !$ symbol;
- know how to negate a statement of the form "$\exists ! x \; (P(x))$";
- know how to prove "there exists a unique" statement (Example 3.6.2, all problems from 3.6);
Problems
Turn in: 10.
Extra Problems: 2, 7.
Section 4.1
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand ordered pairs and Cartesian products;
- know how to prove statements involving set operations of Cartesian products (Theorem 4.1.3 and Problems 4.1.9 and 4.1.10);
- know how to find truth sets of statements in more than one variable (Example 4.1.6 and Problem 4.1.3).
Problems
Turn in: 9, 10.
Extra Problems: 3, 7.
Section 4.2
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand what a relation is;
- understand domains, images, inverses and compositions of relations and be able to prove statements involving these.
Problems
Turn in: 5, 8.
Extra Problems: 2, 3, 6(b).
Section 4.3
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand graphic representation of relations (Problem 4.3.5);
- understand relations on a (single) set;
- understand (and be able to identify) reflexive, symmetric and transitive relations (most problems from 4.3).
Problems
Turn in: 14, 16.
Extra Problems: 2, 4, 9, 12, 21.
Section 4.4
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- know what an antisymmetric relation is;
- know what are total and partial orders (and their difference -- Problem 4.4.2);
- know what least/greatest (or smallest/largest) and minimal/maximal elements are (and their difference) (Examples 4.4.5, 4.4.7, Problem 4.4.3);
- know what upper/lower bound are and what greatest lower bound and least upper bound are (Example 4.4.10, Problem 4.4.3).
Problems
Turn in: 6, 22.
Extra Problems: 2, 3, 9, 15.
Section 4.6
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- know what equivalence relation and equivalency class are;
- know what a partition is;
- know the relation between partitions and equivalence relations;
- know what are congruences modulo $n$ (as in $a \equiv b \pmod{n}$).
Problems
Turn in: 13, 20.
Extra Problems: 4, 8, 16.
Section 5.1
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand functions as relations (and be able to prove basic facts about them).
Problems
Turn in: 9(b), 17(b).
Extra Problems: 9(a), 11, 13, 17(a).
Section 5.2
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- know the definitions of one-to-one/injective and onto/surjective functions;
- know how to prove statements involving one-to-one and onto functions, including those involving compositions.
Problems
Turn in: 8(b), 9(a).
Extra Problems: 3, 6, 11, 18.
Section 5.3
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand the definition of the inverse function;
- know the necessary and sufficient conditions to the existence of the inverse (e.g., Theorems 5.3.4-5).
Problems
Turn in: 10, 12.
Extra Problems: 4, 6.
Section 6.1
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand the process of induction and how/why it works;
- be able to use induction to prove formulas (Example 6.1.1, Problems 6.1.4, 6.1.16), inequalities (Example 6.13, Problem 6.1.14) and divisibility questions (Example 6.1.2, Problem 6.1.9).
Problems
Turn in: 9(b), 16.
Extra Problems: 4, 9(a).
Section 6.2
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- be able to prove by induction statements that are not formulas, such as statements on finite sets (Examples 6.2.1, 6.2.2, Problem 6.2.3), combinatotics (Problem 6.2.10) and geometry (Examples 6.2.3, 6.2.4).
Problems
Turn in: 3, 6 (here you can use, without proving, the Triangle Inequality: if $a, b \in \mathbb{R}$, then $|a+b| \leq |a|+|b|$).
Extra Problems: 5, 10.
Section 6.3
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand recurrence and recursive definitions;
- be able to apply induction to prove summation formulas (Problems 6.3.2, 6.3.5, 6.3.9);
- be able to apply induction to prove formulas recursively defined sequences (Example 6.3.3, Problem 6.3.16);
- be able to apply induction to more complex inequalities (for integers -- Example 6.3.1, Problem 6.3.12).
Problems
Turn in: None.
Extra Problems: 2, 5, 9, 12, 16.
Section 6.4
Videos
Outcomes
After the assignment (reading and videos before class) and class, you should:
- understand and know how to use strong induction (Theorem 6.4.2);
- understand more complex examples of recursively defined sequences (Problem 6.4.4);
- know what Fibonacci numbers are and be able to prove simple statements about them (Theorem 6.4.3, Problems 6.4.6, 6.4.7).
Problems
Turn in: None.
Extra Problems: 4, 6, 7, 19.