Work These Problems Again This Time Using Brief Truth Tables
Truth Tables
Considering complex Boolean statements tin get tricky to recollect about, we can create a truth table to proceed track of what truth values for the simple statements make the complex argument truthful and false
Truth Table
A tabular array showing what the resulting truth value of a complex statement is for all the possible truth values for the simple statements.
Example 1
Suppose you're picking out a new couch, and your significant other says "go a sectional or something with a chaise."
This is a complex statement made of two simpler conditions: "is a sectional," and "has a chaise." For simplicity, allow's utilize Due south to designate "is a sectional," and C to designate "has a chaise." The condition S is true if the burrow is a sectional.
A truth table for this would look similar this:
| S | 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 first row, if S is truthful and C is also truthful, then the complex statement "Southward or C" is true. This would be a sectional that too has a chaise, which meets our desire.
Call up also that or in logic is not sectional; if the burrow has both features, information technology does meet the condition.
To shorthand our notation farther, we're going to introduce some symbols that are commonly used for and, or, and not.
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 non: not A is notated ~A
You lot can remember the beginning 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. Besides, A ⋁ B would be the elements that exist in either set, in A ⋃ B.
In the previous example, the truth table was really just summarizing what nosotros already know almost how the or argument 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 become useful when analyzing more circuitous Boolean statements.
Example ii
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. We start by list all the possible truth value combinations for A, B, and C. Detect how the beginning column contains 4 Ts followed by 4 Fs, the second column contains two Ts, two Fs, then repeats, and the last column alternates. This blueprint ensures that all combinations are considered. Along 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 |
Next we can find the negation of B ⋁ C, working off the B ⋁ C column nosotros 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 observe 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 |
Information technology turns out that this circuitous expression is merely truthful in one case: if A is truthful, B is false, and C is imitation.
When we discussed weather condition earlier, we discussed the type where we accept an action based on the value of the status. We are now going to talk about a more general version of a conditional, sometimes called an implication.
Implications
Implications are logical conditional sentences stating that a argument p, called the ancestor, implies a consequence q.
Implications are commonly written as p → q
Implications are similar to the conditional statements we looked at earlier; p → q is typically written equally "if p and then q," or "p therefore q." The divergence between implications and conditionals is that conditionals we discussed earlier suggest an action—if the status is truthful, and so we take some action as a result. Implications are a logical statement that propose that the consequence must logically follow if the antecedent is true.
Example 3
The English language statement "If it is raining, then there are clouds in the sky" is a logical implication. It is a valid statement because if the antecedent "information technology is raining" is truthful, and then the event "there are clouds in the sky" must also be true.
Notice that the statement tells u.s.a. cipher of what to expect if it is not raining. If the antecedent is simulated, then the implication becomes irrelevant.
Case iv
A friend tells yous that "if yous upload that pic to Facebook, you'll lose your job." In that location are four possible outcomes:
- You upload the film and keep your job
- You lot upload the picture show and lose your chore
- Y'all don't upload the picture and keep your task
- Yous don't upload the moving picture and lose your chore
There is only i possible case where your friend was lying—the first option where you lot upload the picture and go on your job. In the last two cases, your friend didn't say annihilation about what would happen if you lot didn't upload the picture, so y'all can't conclude their statement is invalid, even if you didn't upload the motion-picture show and all the same lost your job.
In traditional logic, an implication is considered valid (truthful) as long equally there are no cases in which the antecedent is truthful and the consequence is faux. It is important to proceed 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 |
Example v
Construct a truth tabular array for the statement (one thousand ⋀ ~p) → r
We start by constructing a truth table for the antecedent.
| one thousand | p | ~p | m ⋀ ~p |
| T | T | F | F |
| T | F | T | T |
| F | T | F | F |
| F | F | T | F |
At present we can build the truth table for the implication
| m | p | ~p | g ⋀ ~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 example, when k is true, p is false, and r is fake, then the antecedent m ⋀ ~p will be true but the issue imitation, resulting in a invalid implication; every other instance gives a valid implication.
For whatsoever implication, at that place are three related statements, the converse, the inverse, and the contrapositive.
Related Statements
The original implication is "if p so q": p → q
The converse is "if q so p": q → p
The changed is "if not p then not q": ~p → ~q
The contrapositive is "if not q then non p": ~q → ~p
Example 6
Consider again the valid implication "If it is raining, and so there are clouds in the sky."
The antipodal would be "If there are clouds in the sky, information technology is raining." This is certainly not always truthful.
The inverse would exist "If information technology is non raining, then at that place are not clouds in the heaven." Likewise, this is not e'er true.
The contrapositive would be "If there are not clouds in the sky, then information technology is non raining." This argument is valid, and is equivalent to the original implication.
Looking at truth tables, nosotros can run across that the original conditional and the contrapositive are logically equivalent, and that the converse and inverse are logically equivalent.
| Implication | Converse | 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 conditional statement and its contrapositive are logically equivalent.
The converse and inverse of a statement are logically equivalent.
Arguments
A logical argument is a claim that a gear up of bounds support a conclusion. There are ii full general types of arguments: inductive and deductive arguments.
Argument types
An anterior argument uses a collection of specific examples every bit its premises and uses them to advise a general decision.
A deductive argument uses a collection of general statements equally its premises and uses them to propose a specific situation equally the conclusion.
Example 7
The statement "when I went to the store last week I forgot my purse, and when I went today I forgot my purse. I ever forget my bag when I become the shop" is an inductive statement.
The premises are:
I forgot my purse final week
I forgot my handbag today
The conclusion is:
I ever forget my purse
Discover that the premises are specific situations, while the determination is a general argument. In this case, this is a adequately weak argument, since it is based on merely two instances.
Example 8
The argument "every 24-hour interval for the past yr, a plane flies over my house at 2pm. A aeroplane will fly over my house every twenty-four hours at 2pm" is a stronger inductive argument, since it is based on a larger gear up of evidence.
Evaluating inductive arguments
An anterior statement is never able to prove the conclusion true, merely it 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 diversity 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, similar Newton's theory of gravity, have all stood up to years of testing and testify, though sometimes they need to exist adjusted based on new evidence. For gravity, this happened when Einstein proposed the theory of general relativity.
A deductive argument is more clearly valid or not, which makes them easier to evaluate.
Evaluating deductive arguments
A deductive statement is considered valid if all the premises are true, and the conclusion follows logically from those premises. In other words, the premises are truthful, and the determination follows necessarily from those premises.
Example 9
The argument "All cats are mammals and a tiger is a cat, then a tiger is a mammal" is a valid deductive argument.
The premises are:
All cats are mammals
A tiger is a cat
The conclusion is:
A tiger is a mammal
Both the premises are truthful. To see that the premises must logically lead to the conclusion, one approach would be use a Venn diagram. From the showtime premise, we tin can conclude that the gear up of cats is a subset of the set of mammals. From the 2nd premise, nosotros are told that a tiger lies inside the set of cats. From that, we can run into in the Venn diagram that the tiger likewise lies inside the ready of mammals, so the determination is valid.
Analyzing Arguments with Venn Diagrams[ane]
To analyze an argument with a Venn diagram
- Draw a Venn diagram based on the premises of the argument
- If the premises are insufficient to determine what determine the location of an element, betoken that.
- The argument is valid if it is articulate that the conclusion must be truthful
Example 10
Premise: All firefighters know CPR
Premise: Jill knows CPR
Decision: Jill is a firefighter
From the offset 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 fellow member of that larger set, merely we do not take enough information to know if she also is a member of the smaller subset that is firefighters.
Since the conclusion does non necessarily follow from the premises, this is an invalid argument, regardless of whether Jill really is a firefighter.
It is of import to note that whether or not Jill is actually a firewoman is not important in evaluating the validity of the argument; we are only concerned with whether the bounds are enough to prove the conclusion.
In addition to these chiselled way premises of the form "all ___," "some ____," and "no ____," it is also common to see premises that are implications.
Example xi
Premise: If you live in Seattle, you live in Washington.
Premise: Marcus does non live in Seattle
Conclusion: Marcus does non live in Washington
From the first premise, nosotros know that the set of people who live in Seattle is inside the set of those who alive in Washington. From the second premise, we know that Marcus does not lie in the Seattle fix, just nosotros have insufficient information to know whether or non Marcus lives in Washington or not. This is an invalid argument.
Example 12
Consider the argument "You lot are a married man, then you must take a married woman."
This is an invalid argument, since in that location are, at least in parts of the world, men who are married to other men, so the premise not insufficient to imply the conclusion.
Some arguments are better analyzed using truth tables.
Case thirteen
Consider the argument:
Premise: If you bought bread, then you went to the store
Premise: You bought bread
Conclusion: You went to the shop
While this instance is hopefully fairly obviously a valid statement, we can clarify it using a truth table past representing each of the premises symbolically. We can then wait at the implication that the bounds together imply the conclusion. If the truth table is a tautology (always true), then the statement is valid.
Nosotros'll get B represent "yous bought bread" and Southward represent "you went to the store". Then the argument becomes:
Premise: B → Due south
Premise: B
Decision: S
To test the validity, we await at whether the combination of both bounds implies the conclusion; is it true that [(B→S) ⋀ B] → South ?
| B | S | B → S | (B→S) ⋀ B | [(B→S) ⋀ B] → South |
| 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 truthful, this is a valid argument.
Analyzing arguments using truth tables
To analyze an argument with a truth table:
- Stand for each of the premises symbolically
- Create a conditional statement, joining all the bounds with and to form the antecedent, and using the conclusion as the consequent.
- Create a truth table for that argument. If it is ever true, and then the statement is valid.
Example 14
Premise: If I become to the mall, then I'll buy new jeans
Premise: If I buy new jeans, I'll buy a shirt to get with it
Determination: If I got to the mall, I'll purchase a shirt.
Let Thousand = I get to the mall, J = I buy jeans, and S = I buy a shirt.
The premises and determination can exist stated as:
Premise:M → J
Premise:J → Due south
Conclusion: 1000 → S
Nosotros tin construct a truth table for [(1000→J) ⋀ (J→S)] → (Yard→Due south)
| M | J | South | Grand → J | J → S | (M→J) ⋀ (J→S) | M → S | [(Chiliad→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, we tin can see this is a valid statement.
Source: https://courses.lumenlearning.com/math4libarts/chapter/truth-tables-and-analyzing-arguments-examples/
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