**Non-linearity in a Network**

A network is said to be non-linear if it does not satisfy both the homogeneity and superposition conditions. Let's explore these conditions in more detail:

**Homogeneity Condition:**

The homogeneity condition states that if the input to a network is scaled by a constant factor, then the output of the network should also be scaled by the same factor.

Mathematically, for a linear network:

f(a * x) = a * f(x)

where 'f' represents the network function, 'x' represents the input to the network, and 'a' represents a constant scaling factor.

If a network does not satisfy this condition, it means that the output of the network does not scale proportionally to the input when multiplied by a constant factor. This violates the homogeneity condition and indicates non-linearity.

**Superposition Condition:**

The superposition condition states that the response of a network to a sum of inputs is equal to the sum of the responses of the network to each individual input.

Mathematically, for a linear network:

f(x1 + x2) = f(x1) + f(x2)

where 'f' represents the network function, 'x1' and 'x2' represent two different inputs to the network.

If a network does not satisfy this condition, it means that the output of the network to a sum of inputs is not equal to the sum of the responses to each individual input. This violates the superposition condition and indicates non-linearity.

**Non-Linearity:**

If a network does not satisfy both the homogeneity and superposition conditions, it is considered non-linear. This means that the network's response is not solely determined by the input and exhibits complex behavior that cannot be represented by simple scaling or superposition of inputs.

Non-linear networks are commonly found in various fields of study, including electrical engineering, physics, and biology. They are often characterized by non-linear equations, complex interactions, and emergent behavior.

In electrical engineering, non-linear networks play a significant role in modeling and analyzing systems that exhibit non-linear behavior, such as diodes, transistors, and power electronic converters. Understanding non-linearity is crucial for designing accurate and efficient systems in these fields.

Therefore, the correct answer to the question is option 'C' - both the homogeneity and superposition conditions need to be violated for a network to be considered non-linear.