# Logic Unit L.0 Quadratic Formula, Painted Cube Activity Worksheet

```Logic Unit
L.0 Quadratic Formula, Painted Cube Activity
Worksheet D#1-17
ax^2+bx+c = 0
 b  b 2  4ac
x
2a
- work in teams on 3 pages related to cubes
- homework is also to finish what you don’t get done in class
L.1 Introduction to Logic
Part 1 exercises (all)
- Logic – the study of arguments
- we will learn how to analyze existing statements in a
rational way and be able to deduce new ones
- you might have seen this on Star Trek, Sherlock Holmes, debate,
famous philosophers like Plato and Confucious,….
- simple statement – a statement that is either T or F and contains no
other statement inside it
- can assign a variable to a statement: c
- “I like the Colts”
~c means “I don’t like them
- V = means OR (disjunction)
- ^ (upside down V) means AND (conjunction)
- can form complex statements with AND/OR
- (p V q) is true if one of them is true
- (p ^ q) is true if both are true
- do exs where are given p,q as statements and then come up with
English for complex statements
- p= Peyton Manning is great
- q = Dolphins will go to the Playoffs
Ex: ~q, p ^ q, ~ (p ^ q),…
- you can make a truth table to figure out the value of a complex
statement based on all possible combinations of the values of simple
statement
- do a table with p, q, (p V q), (p ^ q), ~(p ^ q), ~(p V q)
- do problems like page 5 where you give statements a value and then
make sentences with them
L.2 Logical Equivalence and Laws of Logic
Part 2 all
- logically equivalent statements – two statements that have the same
truth/false values for all combinations of simple statement values
- symbol of equivalence is 
- ex: p  ~(~p)
- ex: p ^ q  q ^ p
- ex: ~(p V q)  (~p) ^ (~q)
- explain in English
- important equivalence statements:
- Double Negative (above)
- Commutative Property (for ^, V)
- DeMorgan’s Law: ~(p V q)  ~p ^ ~q
and also ~(p ^ q)  ( ~p) V (~q)
- distributing the “not” flips the conj/disj
- Associative Property: p V (q V r)  (p V q) V r
- same for conjunction
- Distributive Property: p ^ (q V r)  (p ^ q) V (p ^ r)
- and also p V (q ^ r)  (p V q) ^ (p V r)
- Absorption: p ^ p  p, p V p  p
Ex: do ~(p ^ (~q))  ….. ~p V q [use DeMorgan’s Law
- tautology – a statement that is always true for all values of simple
statements
- ex: p V ~p
- contradiction – a statement that is always false for all values of simple
statements
- ex: p ^ ~p
- is the following a contradiction? p ^ ~(p V q)? (yes)
- can use Truth tables to prove any tautology
- for every combination of simple statements, find both sides of the
 and make sure they are the same
L.3 The Conditional
Part 3 all
- conditional – statement in the form if ___ then __
- written as p -&gt; Q
- pronounced: “p implies Q”
- also as “if p, then q”
- p is called the hypothesis/premise/given
- q is the conclusion
- truth table describes how p -&gt; q works (tricky)
P
Q
P -&gt; Q
F
F
T
F
T
T
T
F
F
T
T
T
- an implication is only false if the hypothesis is True and conc is
False
- so if the hypothesis is false, everything is fine and the implication
is true (not necessarily the conclusion)
- are p-&gt;q  ~p V q ?
- yes since false p is success, other wise q is T…so success
- this is an important rule...allows you to switch between
implication and disjunctive forms
- this is called Equivalent Disjunctive Form (EDF)
- given that B is true, what can be said of A -&gt; B?
- it has to be true since it can only be false if B is false
- biconditional – means “if and only if” (iff) – a double implication
where each side implies the other
- so A &lt; - &gt; B means A- &gt;B and B-&gt;A
- write out the Truth table on p8
- has implication, converse, inverse, contrapositive, and
biconditional
-hw has a lot of truth tables
L.4 Five More Laws of Logic
- L.4 exs #1, 3-5 (so skip #2)
- can use -&gt; to create some new tautologies
1. Simplification: (p ^ q) -&gt; q
2. Law of Syllogism (Transitivity): [(p -&gt; q) ^ (q-&gt; r)] -&gt; (p-&gt; r)
- can chain this as much as you like
3. Disjunctive Syllogism: [(p V q) ^ (~p)] -&gt; q
4. Modus Ponens (direct reasoning): [(p-&gt;q) ^ p]-&gt; q
5. Modus Tollens (indirect reasoning): [(p-&gt;q) ^ ~q] -&gt; ~p
- notice how it involved the contrapositive
- do examples from p14 where need to determine if it’s MP, MT, or
neither
- if P is necessary for Q, then Q-&gt;P [not assigning these as homework]
- if X is sufficient for Y, then X-&gt; Y
L.5 Logic Proofs
- part 5 all except for #3 (skip it)
- premise – statement that we either are given or prove to be true
- inference – a maneuver that results in a new premise being made from
old ones
- argument – list of statements and the conclusion
- proof – argument and list of reasons for each step
- do 3 problems on p18
- when doing homework, it helps to start out thinking about the
conclusion when you are planning strategy
L.6 Sets
- 2 days
part 5 all
- see the packet for notes
```