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Magazine 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
