2. Prove the second distributive law:

[tex]m(\overline{A} + \overline{B}) = m\overline{A} + m\overline{B}[/tex]

3. Define the modulus of a vector.

The second distributive law states that m ( A + B ) = m A + m B , which can be proven by distributing 'm' to both terms. The modulus of a vector represents its length and is calculated differently in two and three dimensions based on its components. Understanding these concepts is fundamental in mathematics and physics.
;

Answered by Anonymous | 2025-07-04

Proving the second distributive law :

The second distributive law in Boolean algebra states:
m ( A + B ) = m A + m B
To prove this law, we can use the properties of Boolean algebra. Here's a step-by-step proof:

We assume A and B are Boolean variables that can take values 0 or 1.

The operation + represents the OR operation, and m is taken as a multiplier.

Using the distributive property in arithmetic, the expression expands as follows:


m ( A + B ) = m A + m B

This expression shows that if either A or B is true, then modifying (using m ) each individually leads to the same result.

This logical property aligns with basic principles of algebra extended into Boolean contexts.

Defining the modulus of a vector :

The modulus (or magnitude) of a vector is a measure of its length.
Given a vector v = ⟨ x , y , z ⟩ in a three-dimensional space, its modulus is calculated using the formula:
∥ v ∥ = x 2 + y 2 + z 2 ​
This formula essentially uses the Pythagorean theorem in three dimensions to determine the length of the vector from the origin point (0, 0, 0) to the coordinates ( x , y , z ) .
In simple terms, the modulus gives you a scalar value representing how far the vector reaches from the origin, regardless of its direction.

Answered by AvaCharlotteMiller | 2025-07-06