Logic A statement is a sentence that is either true or false (its truth value). Logically speaking,...
-
Upload
cory-patrick -
Category
Documents
-
view
215 -
download
0
Transcript of Logic A statement is a sentence that is either true or false (its truth value). Logically speaking,...
![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/1.jpg)
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.
![Page 2: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/2.jpg)
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.
![Page 3: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/3.jpg)
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.
![Page 4: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/4.jpg)
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.
notnot
![Page 5: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/5.jpg)
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.
![Page 6: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/6.jpg)
Converse
A statement’s converse will switch its hypothesis and conclusion.
Statement:If I am happy, then I smile.
Converse:If , then .I smileI am happy
![Page 7: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/7.jpg)
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.
![Page 8: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/8.jpg)
ContrapositiveA 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 .I am not awake
my alarm has not gone off not
not
![Page 9: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/9.jpg)
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!
![Page 10: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/10.jpg)
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)
q
p
or
or
~
![Page 11: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/11.jpg)
Examples
If p, then q Inverse:
If not p, then not q Converse:
If q, then p Contrapositive
If not q, then not p
qp
qp ~~
pq
pq ~~
![Page 12: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/12.jpg)
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
TF
F
![Page 13: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/13.jpg)
A conditional truth table
p q p → q
T T
T F
F T
F F
T
F
T
T
![Page 14: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/14.jpg)
A conditional truth table
p q p → q
T T
T F
F T
F F
T
F
T
T
q → p ~p →~q ~q →~p
T
T
F
T
T
T
F
T
T
F
T
T
![Page 15: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/15.jpg)
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.
Logical Equivalents
![Page 16: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/16.jpg)
Logical Equivalents
Which of the following statements are logically equivalent?
p q p → q
T T
T F
F T
F F
T
F
T
T
q → p ~p →~q ~q →~p
T
T
F
T
T
T
F
T
T
F
T
T
![Page 17: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/17.jpg)
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 qp
![Page 18: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/18.jpg)
Conjunctions A conjunction is true only if
both statements are true.
p q p ^ q
T T
T
T
T
F
F
F F
F
F
F
Remember: the truth value of a conjunction refers to the statement as a whole.
Consider: “The sun is out and it is raining.”
![Page 19: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/19.jpg)
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 qp
![Page 20: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/20.jpg)
Disjunctions A disjunction is true if either
statement is true.
p q p v q
T T
T
T
T
F
F
F F
T
T
F
Consider: “Timmy goes to Stanton or he goes to Paxon.”
![Page 21: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649dcf5503460f94ac3ad0/html5/thumbnails/21.jpg)
Homework
Page 129 #1-14 all