Problem 21 - Discrete Mathematics with Applications

Question

Determine if the given function is invertible. If it is not invertible, explain why. \(f: S \rightarrow \mathbb{N}\) defined by \(f(x)=\) decimal value of \(x,\) where \(S\) is the set of binary representations of positive integers with no leading zeros.

Step-by-Step Solution

Answer

The given function \(f(x)\) is invertible because it is both injective and surjective. It is injective as distinct binary representations in the domain S are mapped to distinct decimal values, and it is surjective since every positive integer in the codomain N has a unique binary representation in the domain S.

Step 1: Determine Injectivity

To check for the injectivity of the function, we need to show that for any two distinct elements \(x_1\) and \(x_2\) in the domain S, their images under the function f are distinct, i.e., \(f(x_1) \neq f(x_2)\) for all \(x_1 \neq x_2\). So we need to verify if the function assigns distinct decimal values to distinct binary representations. Consider two distinct binary representations in S, say \(x_1\) and \(x_2\). Since \(x_1 \neq x_2\), their decimal values will also be different. Therefore, the function \(f(x)\) will map \(x_1\) to a different decimal value than \(x_2\). Thus, \(f(x_1) \neq f(x_2)\), confirming the injectivity of the function.

Step 2: Determine Surjectivity

To check for the surjectivity of the function, we need to show that for each element \(y\) in the codomain N (the set of positive integers), there exists an element \(x\) in the domain S such that \(f(x) = y\). In other words, we need to verify if every positive integer has a corresponding unique binary representation. Every positive integer can be expressed uniquely in binary representation. For instance, let's take an arbitrary positive integer, \(y\). We can always find its unique binary representation by dividing y by 2 repeatedly, and noting the remainders. These remainders, when read in reverse order, form the unique binary representation of y, say \(x\). Thus, for every positive integer \(y\), there exists a unique binary representation \(x\) in S, such that \(f(x)=y\). This confirms the surjectivity of the function.

Step 3: Conclusion

Since the function \(f(x)\) is both injective and surjective, it is invertible. Therefore, the function that maps binary representations of positive integers to their decimal values is indeed invertible.

Key Concepts

Injectivity

Injectivity refers to a property of a function \(f\) where each element in the function's domain maps to a unique element in the codomain. For a function to be injective, if \(f(x_1) = f(x_2)\) then it must follow that \(x_1 = x_2\).The given function \(f: S \rightarrow N\), where \(S\) is the set of binary representations and \(N\) is the set of positive integers, is said to be injective if distinct binary representations map to distinct decimal numbers. In simpler terms, the function does not produce the same result (decimal value) for two different inputs (binary representations). Consider binary numbers like "101" and "110", which represent the decimal numbers 5 and 6 respectively. Since these outputs are different, the function \(f\) is injective.

Surjectivity

Surjectivity of a function means that every element in the function's codomain (the target set) is an output for some input from the domain. In our function \(f : S \rightarrow N\), surjectivity is confirmed by the fact that every positive integer has a corresponding binary representation. To find the binary representation of a positive integer, perform a series of divisions by 2 and record the remainders. When these remainders are read in reverse, they provide a unique binary number that corresponds to the positive integer. As every positive integer has such a binary representation, it shows that each number in \(N\) has at least one correspondent in \(S\). Thus, the function is surjective.

Binary Representation

Binary representation is a way of expressing numbers in base-2, which uses only two digits: 0 and 1. Every positive integer can be converted into a binary representation by continuously dividing the number by 2, and noting down the remainder at each step. For example, to convert decimal 13 to binary:

• 13 divided by 2 is 6, remainder 1.
• 6 divided by 2 is 3, remainder 0.
• 3 divided by 2 is 1, remainder 1.
• 1 divided by 2 is 0, remainder 1.

Reading the remainders from bottom to top, 13 is represented in binary as "1101". Each binary number corresponds uniquely to a positive integer, confirming the bijective nature which includes injectivity and surjectivity.

Positive Integers

Positive integers, also known as natural numbers, are the set of all whole numbers greater than zero. These numbers include 1, 2, 3, and so on, extending infinitely. When considering the function \(f : S \rightarrow N\), positive integers (in \(N\)) serve as the outputs for the corresponding binary representations (in \(S\)).In a decimal-to-binary conversion, each positive integer can be inverted back into its original form once its binary representation is known fully.This process affirms the invertibility of the function mapping binary numbers to decimal numbers.It further confirms that positive integers effectively fill the codomain, aided by their unique properties of order and sequence, ensuring the function is surjective.