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Chapter 1: Problem 52

Construct a truth table for the logical operator \(N O R .\)

Short Answer

Expert verified
The NOR result is true only when both inputs are false.

Step by step solution

01

- Understand the Inputs

Identify the truth values for the two input variables, A and B. Each variable can either be true (T) or false (F). Write down all possible combinations of these two variables.
02

- Define NOR Operation

Recall that the NOR operation is the negation of the OR operation. The OR operation returns true if at least one input is true, and false otherwise. Therefore, NOR will return true only when both inputs are false, and false in all other cases.
03

- Construct the Table

Create a table with columns for A, B, the intermediate OR result, and the final NOR result. Fill in the four rows corresponding to the input combinations. The rows will be: (A = F, B = F), (A = F, B = T), (A = T, B = F), (A = T, B = T).
04

- Compute OR for Each Row

Calculate the OR operation for each pair of inputs and write down the result in the OR column. The results should be T when either A or B is T, and F when both are F.
05

- Apply NOR Operation

Negate the result of the OR operation to get the NOR result. Write this in the NOR column. Remember, NOR is true only if both inputs are false.
06

- Complete the Truth Table

Review and verify that all entries are correct. Here is the completed truth table: | A | B | A OR B | A NOR B ||---|---|--------|---------|| F | F | F | T || F | T | T | F || T | F | T | F || T | T | T | F |

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Key Concepts

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

Logical Operators
Logical operators form the basis of logical reasoning in many areas, including mathematics, computer science, and digital electronics. They are used to perform logical operations on variables and return specific outcomes. For example, some common logical operators include AND, OR, and NOR. These operators help to build complex logical expressions. Logical operators can take one or more truth values (either true or false) and compute a result.
NOR Operation
The NOR operation is a fundamental logical operator that combines the operations of OR and NOT (negation). When performing a NOR operation, start with the OR operation, which returns true if at least one of its inputs is true. Next, apply negation to the result. This means NOR only returns true when both inputs are false. To put it simply, the NOR operator first checks if any inputs are true (like OR) and then negates the result. It is denoted as: \[ A \text{ NOR } B = eg (A \text{ OR } B) \]
Truth Values
In logic, we work with truth values, which represent the truthfulness of a statement. Each statement can either be true (T) or false (F). Truth values are the foundation of constructing truth tables and performing logical operations. When working with logical operators like NOR, understanding how to manipulate truth values is essential. The combinations for two variables, A and B, give us the following possible pairs:
  • A = F, B = F
  • A = F, B = T
  • A = T, B = F
  • A = T, B = T
Negation of OR
To grasp the NOR operation fully, it's crucial to understand the concept of negation of OR. The OR operator provides a true result when at least one input is true. For the negation (NOT) part, we simply reverse the OR result. If the OR result is true, negation makes it false, and if the OR result is false, negation makes it true. This interaction is what gives NOR its unique behavior. Thus, NOR is true only when the OR result is false, or in other words, when both inputs are false. Let's see the table summarizing this:
  • When A = F and B = F, A OR B = F, hence A NOR B = T
  • When A = F and B = T, A OR B = T, hence A NOR B = F
  • When A = T and B = F, A OR B = T, hence A NOR B = F
  • When A = T and B = T, A OR B = T, hence A NOR B = F

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Most popular questions from this chapter

a) Show that \(\forall x P(x) \wedge \exists x Q(x)\) is logically equivalent to \(\forall x \exists y(P(x) \wedge Q(y)),\) where all quantifiers have the same nonempty domain. b) Show that \(\forall x P(x) \vee \exists x Q(x)\) is equivalent to \(\forall x \exists y\) \((P(x) \vee Q(y)),\) where all quantifiers have the same nonempty domain.

Determine whether these are valid arguments. a) If \(x\) is a positive real number, then \(x^{2}\) is a positive real number. Therefore, if \(a^{2}\) is positive, where \(a\) is a real number, then \(a\) is a positive real number. b) If \(x^{2} \neq 0,\) where \(x\) is a real number, then \(x \neq 0 .\) Let \(a\) be a real number with \(a^{2} \neq 0 ;\) then \(a \neq 0\)

For each of these arguments, explain which rules of inference are used for each step. a) 鈥淟inda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket.鈥� b) 鈥淓ach of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five roommates can take a course in algorithms next year.鈥� c) 鈥淎ll movies produced by John Sayles are wonder-ful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners.鈥� d) 鈥淭here is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre.鈥�

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. $$ \begin{array}{l}{\text { a) } \forall x\left(x^{2} \geq x\right)} \\ {\text { b) } \forall x(x>0 \vee x<0)} \\ {\text { c) } \forall x(x=1)}\end{array} $$

Express each of these statements using logical operators, predicates, and quantifiers. a) Some propositions are tautologies. b) The negation of a contradiction is a tautology. c) The disjunction of two contingencies can be a tautology. d) The conjunction of two tautologies is a tautology.
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