Problem 12 - Discrete Mathematics with Applications

Question

Let \(f, g: \mathbf{W} \rightarrow \mathbf{W}\) defined by \(f(x)=x \bmod 5\) and \(g(x)=x \operatorname{div} 7 .\) Evaluate each. $$(f \circ g)(78)$$

Step-by-Step Solution

Answer

\( (f \circ g)(78) = 1 \)

Step 1: Evaluate g(78)

Begin by evaluating g(78), using the definition of g(x) = x div 7 :\[ g(78) = 78 \,\text{div}\, 7. \]To divide 78 by 7, we compute:\[ 78 \,\text{div}\, 7 = 11. \]So, \(g(78) = 11\).

Step 2: Evaluate f(g(78))

Now, to evaluate \((f \circ g)(78)\), we'll use the result we found for g(78) as the input to the function f. Recall that f(x) = x % 5, so we have:\[ (f \circ g)(78) = f(g(78)) = f(11). \]Next, we'll compute f(11) using the definition of f(x):\[ f(11) = 11 \bmod 5. \]Using the modulus operation, we determine the remainder when 11 is divided by 5:\[ 11 \bmod 5 = 1. \]Thus, \(f(11) = 1\).

Step 3: Final Answer

We have found that \(f(11) = 1\), and since \((f \circ g)(78) = f(g(78)) = f(11)\), it follows that:\[ (f \circ g)(78) = 1. \]Therefore, the final answer is \(1\).

Key Concepts

Understanding Modulus Operation

The modulus operation is a fundamental concept in mathematics and computer science. It allows us to find the remainder after division of one number by another. For example, if we write \( a \bmod b \), we are interested in the remainder when \( a \) is divided by \( b \). This is helpful for cycles, like wrapping numbers around a fixed range.
In our exercise, we used the modulus operation in \( f(x) = x \bmod 5 \). For instance, when we calculate \( 11 \bmod 5 \), we divide 11 by 5, which gives a quotient of 2 and a remainder of 1. Therefore, \( 11 \bmod 5 = 1 \), and that's how we arrived at \( f(11) = 1 \).Some highlights of modulus operation:

• It repeats with a fixed cycle, useful in computing time cycles or circular arrays.
• It’s also a key operation in programming for tasks involving constraints within a limit.
• Helps in determining even or odd numbers, if \( a \bmod 2 \) gives 0, then \( a \) is even.

Learning Integer Division

Integer division refers to division in which the quotient is an integer without any fractional part. In simpler terms, we discard the remainder and only consider the whole number value of the division result.
For instance, in the exercise, we used integer division in the function \( g(x) = x \text{ div } 7 \). When we divided 78 by 7, we ignored any decimals and only took the whole number part, resulting in 11. Hence, \( g(78) = 78 \text{ div } 7 = 11 \).Here are some key aspects of integer division:

• It’s commonly used in programming, especially for index calculations and array navigation.
• Unlike regular division, the result doesn't include any fractional portion.
• Helps in applications where only whole units matter, like counting items or people.

Exploring Function Evaluation

In mathematics and programming, evaluating a function is about finding its output for a particular input. This involves understanding the rules that define the function and following them to get the result.
In our example exercise, we have two functions involved in a composition. Function composition, like \((f \circ g)(x)\), means you first apply function \( g \) to \( x \), and then apply function \( f \) to the result of \( g(x) \).
For instance:- First, we evaluate \( g(78) \) to get 11.- Then, we find \( f(11) \) which gives us the final result.Key insights into function evaluation include:

• Function composition is akin to passing a function's output as input to another, creating a new functionality.
• It's essential in building complex algorithms, as it allows chaining simple functions for complex computations.
• Understanding the specifics of each function helps in accurately predicting the overall result.