Logic

1
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 2pr.
• Statements can be joined together in various ways
  to make new statements.

2
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.

3
Conditional Statements

• Conditional statements dont 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.

4
Inverse

• The inverse of a statement is formed by negating
  both its hypothesis and conclusion.
• Statement
• If I take out my cell phone, then Mr. Peterson
  will confiscate it.
• Inverse
• If I do take out my cell phone, then Mr.
  Peterson will confiscate it.

not
not
5
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.

6
Converse

• A statements converse will switch its hypothesis
  and conclusion.
• Statement
• If I am happy, then I smile.
• Converse
• If , then .

I smile
I am happy
7
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 Im thirsty, I drink water.
• All rectangles have four right angles.
• If a triangle is isosceles, then two of its sides
  are the same.

8
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
  ,then .

my alarm has not gone off
not

I am not awake
not

9
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!

10
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
• a statement, usually a premise a statement,
  usually a conclusion creates a conditional
  statement negates a statement (takes its
  opposite)

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Examples

• If p, then q
• InverseIf not p, then not q
• ConverseIf q, then p
• ContrapositiveIf not q, then not p

12
Truth Table

• A truth table is a way to organize the truth
  values of various statements.
• In a 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
T
F
13
A conditional truth table
p q p ? q




T
T
T
T
F
F
F
T
T
F
F
T
14
A conditional truth table
p q p ? q




q ? p p ?q q ?p




T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
T
T
T
T
15
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.

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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
F
T
F
T
F
F
T
T
F
F
T
T
T
T
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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

18
Conjunctions

• A conjunction is true only if both statements are
  true.

• Remember the truth value of a conjunction refers
  to the statement as a whole.
• Consider The sun is out and it is raining.

p q p q




T
T
T
T
F
F
T
F
F
F
F
F
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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

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Disjunctions

• A disjunction is true if either statement is true.

• Consider Timmy goes to Stanton or he goes to
  Paxon.

p q p v q




T
T
T
T
F
T
T
F
T
F
F
F
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Homework

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