Great Courses Prove It the Art of Mathematical Argument
Truth Tables
Because complex Boolean statements can become tricky to think about, we tin can create a truth table to keep track of what truth values for the simple statements make the complex statement true and false
Truth Tabular array
A table showing what the resulting truth value of a complex statement is for all the possible truth values for the uncomplicated statements.
Example i
Suppose y'all're picking out a new couch, and your significant other says "get a sectional or something with a chaise."
This is a circuitous statement fabricated of two simpler conditions: "is a exclusive," and "has a chaise." For simplicity, let'south apply S to designate "is a sectional," and C to designate "has a chaise." The condition Due south is true if the burrow is a sectional.
A truth table for this would look similar this:
| South | C | S orC |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
In the table, T is used for true, and F for false. In the start row, if S is true and C is also true, then the complex statement "S or C" is truthful. This would exist a sectional that too has a chaise, which meets our desire.
Remember also that or in logic is not exclusive; if the couch has both features, it does encounter the condition.
To shorthand our notation further, we're going to introduce some symbols that are commonly used for and, or, and non.
Symbols
The symbol ⋀ is used for and: A and B is notated A ⋀ B.
The symbol ⋁ is used for or: A or B is notated A ⋁ B
The symbol ~ is used for not: not A is notated ~A
Y'all can retrieve the first two symbols by relating them to the shapes for the union and intersection. A ⋀ B would be the elements that exist in both sets, in A ⋂ B. Likewise, A ⋁ B would be the elements that be in either ready, in A ⋃ B.
In the previous example, the truth tabular array was really just summarizing what nosotros already know about how the or statement piece of work. The truth tables for the basic and, or, and not statements are shown below.
Basic Truth Tables
| A | B | A ⋀ B |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
| A | B | A ⋁ B |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
| A | ~A |
|---|---|
| T | F |
| F | T |
Truth tables really go useful when analyzing more complex Boolean statements.
Example 2
Create a truth table for the statement A ⋀ ~(B ⋁ C)
It helps to work from the inside out when creating truth tables, and create tables for intermediate operations. Nosotros start by list all the possible truth value combinations for A, B, and C. Notice how the first column contains 4 Ts followed past iv Fs, the second column contains 2 Ts, 2 Fs, and then repeats, and the last column alternates. This blueprint ensures that all combinations are considered. Forth with those initial values, we'll list the truth values for the innermost expression, B ⋁ C.
| A | B | C | B ⋁ C |
| T | T | T | T |
| T | T | F | T |
| T | F | T | T |
| T | F | F | F |
| F | T | T | T |
| F | T | F | T |
| F | F | T | T |
| F | F | F | F |
Side by side nosotros can notice the negation of B ⋁ C, working off the B ⋁ C column we just created.
| A | B | C | B ⋁ C | ~(B ⋁ C) |
| T | T | T | T | F |
| T | T | F | T | F |
| T | F | T | T | F |
| T | F | F | F | T |
| F | T | T | T | F |
| F | T | F | T | F |
| F | F | T | T | F |
| F | F | F | F | T |
Finally, we find the values of A and ~(B ⋁ C)
| A | B | C | B ⋁ C | ~(B ⋁ C) | A ⋀ ~(B ⋁ C) |
| T | T | T | T | F | F |
| T | T | F | T | F | F |
| T | F | T | T | F | F |
| T | F | F | F | T | T |
| F | T | T | T | F | F |
| F | T | F | T | F | F |
| F | F | T | T | F | F |
| F | F | F | F | T | F |
It turns out that this complex expression is only true once: if A is true, B is false, and C is fake.
When we discussed conditions earlier, nosotros discussed the blazon where we accept an action based on the value of the condition. We are now going to talk well-nigh a more general version of a conditional, sometimes called an implication.
Implications
Implications are logical conditional sentences stating that a statement p, chosen the antecedent, implies a consequence q.
Implications are commonly written every bit p → q
Implications are similar to the provisional statements we looked at before; p → q is typically written as "if p then q," or "p therefore q." The difference between implications and conditionals is that conditionals we discussed earlier suggest an action—if the condition is true, then we take some action every bit a event. Implications are a logical statement that propose that the consequence must logically follow if the ancestor is true.
Example 3
The English language argument "If information technology is raining, and then there are clouds in the heaven" is a logical implication. Information technology is a valid statement because if the ancestor "it is raining" is true, then the event "there are clouds in the sky" must also be true.
Notice that the statement tells united states nothing of what to look if it is not raining. If the antecedent is fake, then the implication becomes irrelevant.
Instance 4
A friend tells you that "if yous upload that picture to Facebook, you'll lose your chore." There are four possible outcomes:
- You upload the picture and keep your job
- You upload the flick and lose your chore
- You don't upload the moving-picture show and proceed your job
- You don't upload the picture and lose your task
In that location is only one possible case where your friend was lying—the showtime choice where you upload the picture and keep your task. In the terminal two cases, your friend didn't say anything well-nigh what would happen if y'all didn't upload the picture, so you tin can't conclude their statement is invalid, even if you didn't upload the movie and still lost your job.
In traditional logic, an implication is considered valid (true) as long as there are no cases in which the antecedent is true and the consequence is faux. It is important to keep in mind that symbolic logic cannot capture all the intricacies of the English language.
Truth Values for Implications
| p | q | p → q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Case 5
Construct a truth table for the argument (m ⋀ ~p) → r
We showtime by constructing a truth tabular array for the antecedent.
| thou | p | ~p | chiliad ⋀ ~p |
| T | T | F | F |
| T | F | T | T |
| F | T | F | F |
| F | F | T | F |
Now we can build the truth table for the implication
| m | p | ~p | thou ⋀ ~p | r | (k ⋀ ~p) → r |
| T | T | F | F | T | T |
| T | F | T | T | T | T |
| F | T | F | F | T | T |
| F | F | T | F | T | T |
| T | T | F | F | F | T |
| T | F | T | T | F | F |
| F | T | F | F | F | T |
| F | F | T | F | F | T |
In this case, when m is true, p is false, and r is false, then the antecedent m ⋀ ~p will exist true but the result false, resulting in a invalid implication; every other case gives a valid implication.
For any implication, there are three related statements, the antipodal, the inverse, and the contrapositive.
Related Statements
The original implication is "if p then q": p → q
The antipodal is "if q then p": q → p
The inverse is "if non p then non q": ~p → ~q
The contrapositive is "if not q then not p": ~q → ~p
Case 6
Consider once more the valid implication "If it is raining, then there are clouds in the sky."
The converse would be "If at that place are clouds in the sky, information technology is raining." This is certainly not always true.
The changed would be "If it is not raining, and so there are not clouds in the sky." Likewise, this is not always truthful.
The contrapositive would exist "If in that location are not clouds in the sky, then it is not raining." This argument is valid, and is equivalent to the original implication.
Looking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent, and that the antipodal and changed are logically equivalent.
| Implication | Antipodal | Inverse | Contrapositive | ||
|---|---|---|---|---|---|
| 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 |
Equivalence
A provisional statement and its contrapositive are logically equivalent.
The antipodal and changed of a argument are logically equivalent.
Arguments
A logical argument is a claim that a set of bounds support a conclusion. At that place are ii general types of arguments: inductive and deductive arguments.
Argument types
An inductive argument uses a collection of specific examples as its premises and uses them to suggest a general conclusion.
A deductive argument uses a collection of general statements every bit its premises and uses them to propose a specific situation as the conclusion.
Instance 7
The statement "when I went to the store last week I forgot my handbag, and when I went today I forgot my bag. I always forget my handbag when I go the store" is an inductive argument.
The premises are:
I forgot my pocketbook last calendar week
I forgot my pocketbook today
The conclusion is:
I always forget my purse
Notice that the premises are specific situations, while the conclusion is a general statement. In this case, this is a adequately weak statement, since information technology is based on merely two instances.
Example 8
The argument "every day for the past year, a aeroplane flies over my house at 2pm. A airplane will wing over my business firm every day at 2pm" is a stronger anterior statement, since it is based on a larger set of evidence.
Evaluating inductive arguments
An inductive argument is never able to prove the conclusion true, but information technology can provide either weak or strong evidence to propose it may be true.
Many scientific theories, such as the large blindside theory, can never be proven. Instead, they are inductive arguments supported by a wide diverseness of show. Usually in science, an idea is considered a hypothesis until it has been well tested, at which point it graduates to being considered a theory. The commonly known scientific theories, like Newton's theory of gravity, have all stood upwardly to years of testing and evidence, though sometimes they need to be adapted based on new evidence. For gravity, this happened when Einstein proposed the theory of full general relativity.
A deductive argument is more conspicuously valid or not, which makes them easier to evaluate.
Evaluating deductive arguments
A deductive argument is considered valid if all the premises are true, and the conclusion follows logically from those premises. In other words, the premises are true, and the conclusion follows necessarily from those premises.
Example 9
The argument "All cats are mammals and a tiger is a cat, so a tiger is a mammal" is a valid deductive statement.
The premises are:
All cats are mammals
A tiger is a cat
The conclusion is:
A tiger is a mammal
Both the premises are true. To see that the premises must logically pb to the decision, one arroyo would be apply a Venn diagram. From the starting time premise, nosotros can conclude that the prepare of cats is a subset of the set of mammals. From the second premise, we are told that a tiger lies within the gear up of cats. From that, we can come across in the Venn diagram that the tiger also lies inside the ready of mammals, so the conclusion is valid.
Analyzing Arguments with Venn Diagrams[one]
To clarify an statement with a Venn diagram
- Draw a Venn diagram based on the premises of the statement
- If the premises are bereft to determine what make up one's mind the location of an chemical element, bespeak that.
- The argument is valid if it is clear that the conclusion must be true
Example 10
Premise: All firefighters know CPR
Premise: Jill knows CPR
Determination: Jill is a firefighter
From the starting time premise, we know that firefighters all lie inside the set of those who know CPR. From the second premise, we know that Jill is a member of that larger set up, but we do not have enough information to know if she also is a member of the smaller subset that is firefighters.
Since the conclusion does not necessarily follow from the premises, this is an invalid statement, regardless of whether Jill actually is a firefighter.
It is important to notation that whether or not Jill is actually a fireman is not important in evaluating the validity of the argument; we are but concerned with whether the bounds are enough to prove the conclusion.
In addition to these categorical style premises of the form "all ___," "some ____," and "no ____," information technology is also common to encounter bounds that are implications.
Instance xi
Premise: If y'all alive in Seattle, you live in Washington.
Premise: Marcus does non alive in Seattle
Determination: Marcus does non alive in Washington
From the first premise, we know that the set of people who live in Seattle is within the set of those who live in Washington. From the second premise, we know that Marcus does non lie in the Seattle set, merely we have bereft information to know whether or not Marcus lives in Washington or not. This is an invalid argument.
Example 12
Consider the argument "You are a hubby, so yous must have a wife."
This is an invalid argument, since in that location are, at least in parts of the globe, men who are married to other men, so the premise not insufficient to imply the conclusion.
Some arguments are ameliorate analyzed using truth tables.
Example thirteen
Consider the statement:
Premise: If you bought bread, then you went to the shop
Premise: You bought breadstuff
Conclusion: You went to the store
While this example is hopefully fairly evidently a valid argument, nosotros can analyze it using a truth table by representing each of the premises symbolically. We can so look at the implication that the premises together imply the conclusion. If the truth table is a tautology (always true), then the statement is valid.
We'll go B represent "you bought bread" and S stand for "you went to the store". Then the argument becomes:
Premise: B → Due south
Premise: B
Conclusion: S
To test the validity, we wait at whether the combination of both bounds implies the decision; is it true that [(B→South) ⋀ B] → Southward ?
| B | S | B → S | (B→S) ⋀ B | [(B→S) ⋀ B] → S |
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | F | T |
| F | F | T | F | T |
Since the truth table for [(B→South) ⋀ B] → S is always true, this is a valid argument.
Analyzing arguments using truth tables
To clarify an argument with a truth table:
- Stand for each of the premises symbolically
- Create a conditional statement, joining all the premises with and to grade the ancestor, and using the conclusion as the consequent.
- Create a truth table for that statement. If it is always truthful, then the statement is valid.
Case 14
Premise: If I go to the mall, then I'll purchase new jeans
Premise: If I purchase new jeans, I'll buy a shirt to get with it
Decision: If I got to the mall, I'll buy a shirt.
Let M = I go to the mall, J = I buy jeans, and S = I buy a shirt.
The premises and determination tin can be stated as:
Premise:One thousand → J
Premise:J → S
Decision: Thou → S
We tin construct a truth table for [(M→J) ⋀ (J→S)] → (M→S)
| M | J | S | M → J | J → S | (1000→J) ⋀ (J→S) | Chiliad → S | [(K→J) ⋀ (J→S)] → (M→S) |
| T | T | T | T | T | T | T | T |
| T | T | F | T | F | F | F | T |
| T | F | T | F | T | F | T | T |
| T | F | F | F | T | F | F | T |
| F | T | T | T | T | T | T | T |
| F | T | F | T | F | F | T | T |
| F | F | T | T | T | T | T | T |
| F | F | F | T | T | T | T | T |
From the truth table, nosotros tin can see this is a valid argument.
Source: https://courses.lumenlearning.com/math4libarts/chapter/truth-tables-and-analyzing-arguments-examples/
0 Response to "Great Courses Prove It the Art of Mathematical Argument"
ارسال یک نظر