Operations Associated with Functions

1.7 Operations Associated with Functions

In the world of functions, there are three major properties you’ll often hear about:

1. One-to-One (Injection)
2. Onto (Surjection)
3. Invertible (Bijection)

One-to-One (Injection)

A function \( g: B \to C \) is said to be a one-to-one (injection) function if each element in the domain has a UNIQUE image value. Think of a one-to-one function like a personal locker at the gym. Each person (element in the domain) has their own locker (image value), and no two people share a locker. In other words, every input is linked to a unique output. This is important in cryptography; you don’t want two different messages to generate the same encrypted output.

Onto (Surjection)

Imagine a school bus picking up students and dropping them at various classes. An ‘onto’ function is like making sure every classroom receives at least one student. That is, every output value (codomain) should have at least one corresponding input value. This is crucial in network design to ensure every endpoint can be reached.

Invertible (Bijection)

Invertible functions are the superheroes of the function world. They’re both one-to-one and onto, which means they can be reversed. It’s like having a reversible jacket; you can wear it inside out (inverse), and it still functions as a jacket. In cryptography, this property is essential for secure data encryption and decryption.

Examples

Let’s say we have a function \( h \) defined as:
$$ h = \{(r, 6), (s, 8), (t, 9), (u, 7)\} $$

The inverse of this function would be:

$$ h^{-1} = \{(6, r), (8, s), (9, t), (7, u)\} $$

In the given context, you might have to identify these properties for different functions \( f \), \( g \), \( h \), and \( i \).

• Properties of ( f ): Is it a Function? Is it One-to-One? Is it Onto? Is it Invertible?
• Properties of ( g ): Is it a Function? Is it One-to-One? Is it Onto? Is it Invertible?
• Properties of ( h ): Is it a Function? Is it One-to-One? Is it Onto? Is it Invertible?
• Properties of ( i ): Is it a Function? Is it One-to-One? Is it Onto? Is it Invertible?

Understanding these function properties will help you get a better grasp on things like data mappings, encryption algorithms, and network pathways. It’s fundamental knowledge that you’ll find incredibly useful as you dive deeper into cyberanalysis.