1. Inductive Reasoning
Inductive reasoning is the process of using examples
and observations to reach a conclusion.
Any time you use a pattern to predict what will come
next, you are using inductive reasoning.
A conclusion based on inductive reasoning is called a
conjecture.
2. Counterexamples
A conjecture is either true all of the time, or it is false.
If we wish to demonstrate that a conjecture is true all
the time, we need to prove it through deductive
reasoning.
We will have more on deductive reasoning and the
proof process later. But for now, know that we can
never prove an idea by offering examples that support
the idea.
However, it can be easy to demonstrate that a
conjecture is false. We simply need to provide a
counterexample.
3. Intro to Logic
A statement is a sentence that is either true or false
(its truth value).
Logically speaking, a statement is either true or false.
What are the values of these statements?
The sun is hot.
The moon is made of cheese.
A triangle has three sides.
The area of a circle is 2πr.
Statements can be joined together in various ways to
make new statements.
4. Conditional Statements
A conditional (or propositional) statement has two parts:
A hypothesis (or condition, or premise)
A conclusion (or result)
Many conditional statements are in “If… then…” form.
Ex.: If it is raining outside, then I will get wet.
A conditional statement is made of two separate
statements; each part has a truth value. But the overall
statement has a separate truth value. What are the values
of the following statements?
If today is Friday, then tomorrow is Saturday.
If the sun explodes, then we can live on the moon.
If a figure has four sides, then it is a square.
5. Conditional Statements
Conditional statements don’t have to be “If…
then…” See if you can determine the
condition and conclusion in each of the
following, and restate in “If… then…” form.
An apple a day keeps the doctor away.
What goes up must come down.
All dogs go to heaven.
Triangles have three sides.
6. Inverse
The inverse of a statement is formed by
negating both its premise and conclusion.
Statement:
IfI take out my cell phone, then Mr. Peterson
will confiscate it.
Inverse:
If
I do not take out my cell phone, then Mr.
Peterson will not confiscate it.
7. Try these
Give the inverses for the following
statements. (You may wish to rewrite as
“If… then…” first.) Then determine the truth
value of the inverse.
Barking dogs give me a headache.
If lines are parallel, they will not intersect.
I can use the Pythagorean Theorem on right
triangles.
A square is a four-sided figure.
8. Converse
A statement’s converse will switch its
hypothesis and conclusion.
Statement:
If I am happy, then I smile.
Converse:
If I am happy, then I smile .
9. Try these
Give the converses for the following
statements. Then determine the truth value
of the converse.
If I am a horse, then I have four legs.
When I’m thirsty, I drink water.
All rectangles have four right angles.
If a triangle is isosceles, then two of its sides
are the same.
10. Contrapositive
A contrapositive is a combination of a
converse and an inverse. The premise
and conclusion switch, and both are
negated.
Statement:
If my alarm has gone off,
then I am awake.
Contrapositive:
If my alarm has not gone off,
not
then I am not awakenot .
11. Try these
Give the contrapositives for the following
statements. Then determine its truth value.
If it quacks, then it is a duck.
When Superman touches kryptonite, he gets
sick.
If two figures are congruent, they have the
same shape and size.
A pentagon has five sides.
Note: A contrapositive always has the same
truth value as the original statement!
12. Symbolic representation
Logic is an area of study, related to math (and
computer science and other fields). In formal
logic, we can represent statements symbolically
(using symbols).
Some common symbols:
p a statement, usually a premise
q a statement, usually a conclusion
→ or ⇒ creates a conditional statement
~ or ¬ negates a statement (takes its opposite)
13. Examples
If p, then q p→q
Inverse:
If not p, then not q ~ p →~ q
Converse:
If q, then p
q→ p
Contrapositive
If not q, then not p ~ q →~ p
14. Truth Table
A truth table is a way to organize the truth
values of various statements.
Ina truth table, the columns are statements
and the rows are possible scenarios.
The table contains every possible scenario
and the truth values that would occur.
Example: p ~p
T F
F T
16. A conditional truth table
p q p→q q→p ~p →~q ~q →~p
T T T T T T
T F F T T F
F T T F F T
F F T T T T
17. Logical Equivalents
Two statements are considered logical
equivalents if they have the same truth
value in all scenarios. A way to
determine this is if all the values are the
same in every row in a truth table.
18. Logical Equivalents
Which of the following statements are logically
equivalent?
p q p→q q→p ~p →~q ~q →~p
T T T T T T
T F F T T F
F T T F F T
F F T T T T
19. Conjunctions
A conjunction consists of two statements
connected by ‘and’.
Example:
Water is wet and the sky is blue.
Notation:
A conjunction of p and q is written as p∧q
20. Conjunctions
A conjunction is true only if
both statements are true.
Remember: the truth
p q p
^q value of a conjunction
T T T refers to the statement
as a whole.
T F F
Consider: “The sun is
F T F out and it is raining.”
F F F
21. Disjunctions
A disjunction consists of two statements
connected by ‘or’.
Example:
I can study or I can watch TV.
Notation:
A disjunction of p and q is written as p∨q
22. Disjunctions
A disjunction is true if either
statement is true.
p q pvq Consider: “Timmy
goes to Stanton or he
T T T goes to Paxon.”
T F T
F T T
F F F
23. Biconditional
A biconditional statement is a special type of
conditional statement. It is formed by the conjunction
of a statement and its converse.
Example:
If a quadrilateral has four right angles then it is a rectangle, and
if a quadrilateral is a rectangle then it has four right angles.
Biconditional statements can be shortened by using “if
and only if” (iff.).
A quadrilateral is a rectangle if and only if it has four right
angles.
This is true whether you read it forwards or ‘backwards’.
24. Biconditional
A good definition will consist of a
biconditional statement.
Ex: A figure is a triangle if and only if it has
three sides.
25. Biconditional
A biconditional is true when the
statements have the same truth value.
p q p↔q Consider: “Two distinct
coplanar lines are
T T T parallel if and only if
they have the same
T F F
slope.”
F T F “Our team will win the
playoffs if and only if
F F T
pigs fly.”
26. Venn Diagrams
The truth values of compound statements
can also be represented in Venn diagrams.
p: A figure is a quadrilateral.
q: A figure is convex.
p q
Which part of the diagram
represents:
p∧q
p∧ ~ q
p∨q
~ p∨ ~ q
27. Venn Diagrams – Conditionals
A Venn diagram can represent a conditional
statement:
p: A figure is a quadrilateral.
q: A figure is a square.
p
q