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Chapter 1: Problem 54

Construct a truth table for each proposition. $$p \rightarrow(p \vee q)$$

Short Answer

Expert verified
The truth table for the given proposition \(p \rightarrow(p \vee q)\) is: | \(p\) | \(q\) | \((p \vee q)\) | \(p \rightarrow(p \vee q)\) | |-----|-----|-------------|---------------------------| | T | T | T | T | | T | F | T | T | | F | T | T | T | | F | F | F | T |

Step by step solution

01

List all possible combinations of truth values

We have two propositional variables \(p\) and \(q\). Each of them can be either true (T) or false (F), which gives us a total of 4 possible combinations: 1. T, T 2. T, F 3. F, T 4. F, F
02

Determine truth values for \((p \vee q)\)

In this step, we'll calculate the truth values for the proposition \((p \vee q)\) for each combination of truth values for \(p\) and \(q\). Recall that the disjunction \(p \vee q\) is true if at least one of \(p\) or \(q\) is true and false otherwise. So, we get: 1. T, T -> T 2. T, F -> T 3. F, T -> T 4. F, F -> F
03

Determine truth values for \(p \rightarrow(p \vee q)\)

Now that we have the truth values for \((p \vee q)\), we can calculate the truth values for the main proposition \(p \rightarrow(p \vee q)\). Recall that the conditional \(p \rightarrow q\) is true unless \(p\) is true and \(q\) is false. So, for each combination of truth values for \(p\) and \((p \vee q)\), we get: 1. T -> T -> T 2. T -> T -> T 3. F -> T -> T 4. F -> F -> T
04

Write down the truth table

The truth table for the proposition \(p \rightarrow(p \vee q)\) is as follows: | \(p\) | \(q\) | \((p \vee q)\) | \(p \rightarrow(p \vee q)\) | |-----|-----|-------------|---------------------------| | T | T | T | T | | T | F | T | T | | F | T | T | T | | F | F | F | T | The truth table shows the truth values for the proposition \(p \rightarrow(p \vee q)\) for each possible combination of the truth values for \(p\) and \(q\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Propositional Logic
Propositional logic is a branch of logic that deals with propositions and their relationships. A proposition is a declarative statement that is either true or false, but not both at the same time. This simplicity allows us to focus on the logical relationships between these propositions.
In propositional logic, we use symbols to represent the propositions. For example, 'p' and 'q' may stand for two different statements. We then use logical connectives to build more complex statements, like conjunctions, disjunctions, or conditionals.
  • Each proposition is considered as an entity with a specific truth value.
  • Logical operations help in deriving meaningful conclusions from a set of propositions.
Understanding propositional logic is essential for creating truth tables, which visually represent the truth values of complex propositions.
Conditional Statement
A conditional statement is a logical statement that uses the connective 'if...then...', symbolized as \(p \rightarrow q\). It signifies that if proposition \(p\) (the antecedent) is true, then proposition \(q\) (the consequent) must be true. If the antecedent is false, the conditional statement is simply true, regardless of the truth value of the consequent.
Here are key aspects to remember:
  • It only requires the antecedent to be false, or the consequent to be true, for the statement to be true overall.
  • The only time a conditional statement \(p \rightarrow q\) is false, is when \(p\) is true and \(q\) is false.
This characteristic is crucial for evaluating and understanding truth tables involving conditional statements.
Disjunction
Disjunction is represented by the symbol \(\vee\), and it corresponds to the logical 'OR'. A disjunction \(p \vee q\) is true when at least one of the propositions \(p\) or \(q\) is true. This logical connector allows for more flexibility compared to a conjunction, which demands both propositions to be true.
  • It's like saying either one or both conditions can satisfy the requirement for the disjunction to be true.
  • The disjunction is only false if both individual propositions \(p\) and \(q\) are false simultaneously.
This concept is vital when constructing truth tables, as it lays out scenarios where propositions can be true, based on the defined conditions.
Truth Values
Truth values are integral to understanding and working with propositional logic and truth tables. Each proposition can be either true (T) or false (F). These values form the basis for evaluating logical statements. In a truth table, all possible combinations of truth values for the involved propositions are listed to explore every logical scenario.
Consider the two propositions \(p\) and \(q\):
  • There are four possible combinations of truth values: (T, T), (T, F), (F, T), and (F, F).
  • By systematically evaluating the truth of complex propositions, we can fully understand the implications of logic rules.
Knowing how to work with truth values is fundamental for logic analysis and understanding how different propositions interrelate within truth tables.

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Most popular questions from this chapter

Write the converse, inverse, and contrapositive of each implication. If London is in France, then Paris is in England.

The Sheffer stroke / is a binary operator" defined by the following truth table.(Note: On page 25 we used the vertical bar \(|\) to mean is a factor of. The actual meaning should be clear from the context. So be careful.) Verify each. (Note: Exercise 58 shows that the logical operators \(|\) and \(\mathrm{NAND}\) are the same. (TABLE CAN'T COPY) $$p \rightarrow q \equiv((p | p)|(p | p))|(q | q)$$

Exercises \(65-78\) deal with propositions in fuzzy logic. Let \(p, q,\) and \(r\) be simple propositions with \(t(p)=1, t(q)=0.3,\) and \(t(r)=\) 0.5 . Compute the truth value of each, where \(s^{\prime}\) denotes the negation of the statement \(s\) . $$ \left(p^{\prime}\right)^{\prime} $$

Determine the truth value of each, where \(\mathrm{P}(\mathrm{s})\) denotes an arbitrary predicate. $$(\forall x) P(x) \rightarrow(\exists x) P(x)$$

There are seven lots, 1 through \(7,\) to be developed in a certain city. A builder would like to build one bank, two hotels, and two restaurants on these lots, subject to the following restrictions by the city planning board (The Official LSAT PrepBook, 1991): If lot 2 is used, lot 4 cannot be used. If lot 5 is used, lot 6 cannot be used. The bank can be built only on lot \(5,6,\) or \(7 .\) A hotel cannot be built on lot \(5 .\) A restaurant can be built only on lot \(1,2,3,\) or \(5 .\) If a restaurant is built on lot \(5,\) which of the following is not a possible list of locations? A. A hotel on lot 2 and lot 4 is left undeveloped. B. A restaurant on lot 2 and lot 4 is left undeveloped. C. A hotel on lot 2 and lot 3 is left undeveloped.
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