Question

Write the operation table for the Boolean operation AND.

Short answer

The operation table for AND is: \( (0,0) \to 0; (0,1) \to 0; (1,0) \to 0; (1,1) \to 1 \).

Step-by-step solution

Understand the Boolean Operation AND

The AND operation is a basic binary operation used in Boolean algebra. It is true if and only if both operands are true. We will use binary values 0 and 1 to represent false and true respectively.

Identify the Variables

In this case, we are dealing with two binary variables, which will be the inputs for the AND operation. These variables are typically denoted by A and B.

Formulate Possible Input Combinations

For two binary variables A and B, each can be either 0 (false) or 1 (true). Thus, the total possible combinations are (0,0), (0,1), (1,0), and (1,1).

Apply the AND Operation

Apply the AND operation to each pair of inputs: - If A = 0 and B = 0, the result is 0 (since both are false). - If A = 0 and B = 1, the result is 0 (since A is false). - If A = 1 and B = 0, the result is 0 (since B is false). - If A = 1 and B = 1, the result is 1 (since both are true).

Construct the Operation Table

Based on the results from Step 4, construct the operation table: | A | B | A AND B | |---|---|---------| | 0 | 0 | 0 | | 0 | 1 | 0 | | 1 | 0 | 0 | | 1 | 1 | 1 |

Key concepts

Boolean Operation AND

In Boolean algebra, the AND operation is fundamental. It plays a crucial role in digital circuits and logical problem solving. The AND operation works on two binary inputs. It outputs a true value only when both inputs are true. In this context, a 'true' condition is represented by 1, and 'false' by 0.

Think of the AND operation like a test that both conditions need to pass. Consider two switches connected to a light bulb in series. The bulb will only light up if both switches are flipped to "on" (representing 1). It emphasizes that for the output to be true, every condition must be satisfied.

Binary Variables

Binary variables are the backbone of Boolean algebra. These variables can only take one of two possible values: 0 or 1. In logical operations, these values symbolize the binary states of false and true, respectively.

When working with binary variables, you'll often encounter terms like A, B, C, etc., as placeholders for these variables. They help us in creating logical statements and equations. For example, if variable A stands for a sensor detecting motion, 1 might mean motion is detected, while 0 means no motion.

Operation Table

An operation table is a helpful tool in Boolean algebra. It summarizes the outcomes of a particular operation for all possible inputs. For the AND operation with two binary variables, this table shows every conceivable input combination and their corresponding output.

To construct one, list the binary variables and each possible combination of their values. Then, apply the AND rule to each set of inputs to find the resulting outputs. This systematic approach makes it easier to comprehend how a Boolean function behaves.

Logical Operations

Logical operations like AND are used to perform calculations in binary form. These operations make up the foundation of computer logic and digital circuit design. In addition to AND, boolean algebra includes operations like OR, NOT, NAND, NOR, and XOR, each serving different logical functions.

Understanding these operations helps us build more complex logical statements and solve problems in computational fields. Each operation obeys specific rules, which reflect how different logical conditions interact. Familiarity with these operations is key to mastering logical reasoning in technical domains.