Notes also available as PDF.
Structure of the upcoming test:
Two primary forms of reasoning:
An example of inductive reasoning:
It has been sunny all week, so it will be sunny tomorrow.
There are no explicit rules or assumptions. We just assume the pattern continues.
An example of deductive reasoning:
The weather forecasts state that if the storm turns northward, it will rain tomorrow. The storm has turned northward, so it will rain tomorrow.
This sets up a rule from the weather forecast. Then the rule is applied to data, the storm turning northward, to reach a conclusion about tomorrow.
There rarely are cleancut distinctions. Consider extending the sequence
35,45,55,\mathop{\mathop{…}}

Reasoning inductively, we might assume that the numbers jump by ten. The next number will be 65, then 75, and so on.
Reasoning deductively, we assume this is an arithmetic sequence starting at 35 with an increment of 10. Under this assumption, the next two numbers will be 35 + 10 ⋅ 3 = 65 and 35 + 10 ⋅ 4 = 75.
The difference between the two forms is subtle. Deductive reasoning sets up explicit rules. The rules themselves may be discovered inductively, but making the rules specific and applying them carefully renders the result deductive.
Pólya’s principles:
This is not a simple 1234 recipe. Understanding the problem may include playing with little plans, or trying to carry out a plan may lead you back to trying to understand the problem.
Read the whole problem.
Read all of the problem.
One comment about the homeworks: Most people answer only part of any given problem.
To indicate an answer clearly to someone else (like me), you need to know what the answer is.
Rephrasing the problem may help you remember solution methods.
This is close to devising a plan. Sometimes you may stumble upon an answer.
Examples may help find relationships. The relationships may help you decide on a plan. Mathematics is about relationships between different entities; symbolic mathematics helps abstract away the entities themselves.
Sometimes plans are “trivial,” or so simple it seems pointless to make them specific. But write it out anyways. Often the act of putting a plan into words helps find flaws in the plan.
Try to devise a plan that you can check along the way. The earlier you detect a problem, the easier you can deal with it.
Some plans we’ve considered:
Try a few combinations of the data. See what falls out. This is good for finding relationships and understanding the problem.
If you know the answer lies in some range, you can search that range systematically by building a list.
When trying examples, keep an eye open for patterns. Sometimes the patterns lead directly to a solution, and sometimes they help to break a problem into smaller pieces.
Be sure to understand what results depend on which data. Look for dependencies in the problem. Sometimes pushing the data you have through all the dependencies will break the problem into simpler subproblems.
Attention to detail is critial here.
When building a list, be sure to carry out a welldefined procedure. Or when looking for patterns, be systematic in the examples you try. Don’t jump around randomly.
Can you check your result? Sometimes trying to check reveals new relationships that could lead to a better solution.
Think about how your solution could help with other problems.
A sequence is an ordered list of numbers. Two common kinds are:
One method for extending a sequence is through successive differences. Consider the sequence
11\quad 22\quad 39\quad 64\quad \mathop{\mathop{…}}

To compute the next term, form differences until you find a constant column:
i  {A}_{i}  {Δ}_{i}^{(1)} = {A}_{i}  {A}_{i1}  {Δ}_{i}^{(2)} = {Δ}_{i}^{(1)}  {Δ}_{i1}^{(1)} 
1  11  
2  22  11  
3  39  17  6 
4  62  23  6 
5  91  29  6 
You can write a set by listing its entries, \{1,2,3,4\}, or through set builder notation, \{x\kern 1.66702pt \kern 1.66702pt x\text{ is a positive integer},x < 5\}.
Note that 1 \{1,2\} and \{1\} \{\{1\},\{2\}\}, but \{1\}∉\{1,2\}.
One implication is that ∅⊂ A for all sets A. This statement is vacuously true.
Here \{1\} ⊂\{1,2\} and \{1\}⊄\{\{1\},\{2\}\}.
You can use symbolic logic to write the result of multiple operations.
Some different ways of writing true or false are acceptable:
true  false 
T  F 
1  0 
\mathop{⊤}  ⊥ 
The test’s questions use 1 and 0.
A truth table defining four operations above:
p  q  \mathop{¬}p  p ∧ q  p ∨ q  p → q 
1  1  0  1  1  1 
1  0  0  0  1  0 
0  1  1  0  1  1 
0  0  1  0  0  1 
Note that \mathop{¬}p did not need all four lines. It does not depend on q and has the same value regardless of whether q is false or true.
Consider the truth table:
p  q  f(p,q) 
1  1  1 
1  0  0 
0  1  1 
0  0  0 
We can derive an expression for f(p,q) in two ways. Obviously, any expressions must simplify to q.
One method is to work from the true values. You or together and expressions. There is one and expression per true value. In this case, we have \mathrel{⊧}f(p,q) ≡ (p ∧ q) ∨ (\mathop{¬}p ∧ q). Pulling out the q, this simplifies \mathrel{⊧}f(p,q) ≡ (p ∨\mathop{¬}q) ∧ q ≡ 1 ∧ q ≡ q.
The other method is to work from the false values. You and together or expressions. There is one or expression per false value. Here, \mathrel{⊧}f(p,q) ≡ (\mathop{¬}p ∨ q) ∧ (p ∨ q). Pulling out q again, \mathrel{⊧}f(p,q) ≡ (\mathop{¬}p ∧ p) ∨ q ≡ 0 ∨ q ≡ q.
De Morgan’s laws are two very useful methods for negating terms symbolically:
As an example, consider negating the conditional. Use the equivalent form \mathrel{⊧}p → q ≡\mathop{¬}p ∨ q. Then
So the negation of a conditional is not a conditional itself.
There are four related forms of conditional:
Only the contrapositive has the same meaning as the original conditional, so \mathrel{⊧}p → q ≡\mathop{¬}q →\mathop{¬}p.
The converse and inverse are related to each other but are not equivalent to the original conditional. The inverse is the contrapositive of the converse: \mathrel{⊧}q → p ≡\mathop{¬}p →\mathop{¬}q
The translation of phrases from English to quantified symbolic logic can be tricky.
Almost every student understands all symbolic logic topics.
can translate to
\mathop{∃}s\kern 1.66702pt \mathop{∀}t : \mathop{¬}\mathop{understands}\nolimits (s,t)

because we don’t measure how many but rather that there is or is not one.
Nested quantifiers are not operators. Each quantifier applies to the entire remaining statement. \mathop{∀}s\mathop{∃}t states that for every s, there exists a t for that \mathbf{s}. Meanwhile, \mathop{∃}t\mathop{∀}s states that there exists one single t for every and all s.
Two rules for negating quantifiers:
So saying “not all” is the same as “there exists one for not”, and saying “there does not exist” is the same as “for all, not”.
As an example, we negate the statement above,
Almost every student understands all symbolic logic topics.
and its translation
\mathop{∃}s\kern 1.66702pt \mathop{∀}t : \mathop{¬}\mathop{understands}\nolimits (s,t).

The symbolic negation is
Translating back to English,
All students understand some symbolic logic topic.
It may be true that no two students understand the same topic, but every student understands some topic.