Problem 23 - Discrete Mathematics with Applications

Question

Determine if the functions are bijective. If they are not bijective, explain why. \(f: \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x)=\) decimal value of \(x,\) where \(\Sigma=\\{0,1\\}.\)

Step-by-Step Solution

Answer

The function \(f : \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x) =\) the decimal value of \(x\) is bijective, as it is both injective (each binary string has a unique decimal representation) and surjective (every whole number has a unique binary representation).

Step 1: Analyze function injectivity

To check if the function is injective, we have to verify if every element in the domain (\(\Sigma^{*}\)) has a unique corresponding element in the codomain (\(\mathbf{W}\)). In other words, we need to verify if for any two different elements \(x\) and \(y\) in the domain, their function values are different. In this case, the function \(f(x)\) maps the binary strings to their decimal values, and it is clear that each binary string has a unique decimal representation. Thus, the function is injective.

Step 2: Analyze function surjectivity

To check if the function is surjective, we have to verify if every element in the codomain (\(\mathbf{W}\)) has a corresponding element in the domain (\(\Sigma^{*}\)). In other words, for any whole number, there must be a binary string that represents it. Since we can find a unique binary representation for all whole numbers (even 0 represented by the empty string), the function is surjective.

Step 3: Conclusion

As the function \(f : \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x) =\) the decimal value of \(x\) is both injective and surjective, the function is bijective.

Key Concepts

Injective Functions

Understanding injective functions is crucial to grasp how functions work. An injective function is also known as a "one-to-one" function. This is because each element of the input (or domain) is mapped to a unique element in the output (or codomain). In simpler terms, no two different inputs produce the same output.
For example, if we have a function that maps binary strings to their decimal values, checking for injectivity means ensuring that each binary string maps to a unique decimal number. Since different binary strings will always have different decimal values, the function is injective.
To spot an injective function, remember:

• For every pair of different inputs, say, \(x\) and \(y\), the outputs \(f(x)\) and \(f(y)\) must be different.
• If \(f(x) = f(y)\) implies \(x = y\), then the function is injective.

Injective functions ensure that there are no overlaps in the mapping, making every output distinct.

Surjective Functions

Surjective functions are exciting because they cover the entire codomain. Also known as "onto" functions, they ensure that every possible output value has a corresponding input value. In a surjective function, the aim is that every element in the codomain is mapped by some element in the domain.
For instance, with our function mapping from binary sequences to whole numbers, being surjective means every whole number can be represented as a binary sequence. Thus, for each number in the codomain, there exists at least one string in the domain that it maps from.
Keep these points in mind for surjective functions:

• Every element in the codomain needs to have an originating element in the domain.
• If for every \(w\) in the codomain \(W\), there is at least one \(x\) in \(\Sigma^*\) such that \(f(x) = w\), the function is surjective.

Surjectivity ensures that the function's reach extends to cover all possible outputs.

Binary to Decimal Conversion

Binary to decimal conversion is a fundamental skill, especially in computer science and mathematics. Binary numbers use only 0s and 1s, while decimal numbers use digits from 0 to 9. This conversion helps us understand how computers read and operate numbers.
The conversion process involves understanding place values just like in the decimal system. In a binary number, each digit represents a power of 2, starting from the far right with \(2^0\).
To convert a binary number to decimal, you can follow these steps:

• Write down the binary number and identify each digit's place value (powers of 2).
• Multiply each binary digit by its respective power of 2.
• Add up all the resulting products to get the decimal equivalent.

For example, converting the binary number 1011 involves calculating \(1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 8 + 0 + 2 + 1\), which equals 11 in decimal form.
Understanding this conversion helps in various applications, such as decoding digital signals and implementing binary algorithms.