Problem 28 - Discrete Mathematics with Applications

Question

Let \(f: \mathbf{Z} \times \mathbf{Z} \rightarrow \mathbf{Z}\) defined by \(f(x, y)=2 x+3 y-6 x y .\) Compute the following. $$f(-3,-5)$$

Step-by-Step Solution

Answer

The short answer to the given question is: \(f(-3, -5) = -111 \)

Step 1: Replace the variables x and y with the given values (-3 and -5)

We are given the function \(f(x, y) = 2x + 3y - 6xy\), and we need to compute the value of \(f(-3, -5)\). So, we will replace \(x\) with -3 and \(y\) with -5 in the function: $$ f(-3, -5) = 2(-3) + 3(-5) - 6(-3)(-5) $$

Step 2: Compute the arithmetic operations

Now, we will compute the arithmetic operations: $$ f(-3, -5) = (-6) + (-15) - 6(-3)(-5) $$ Next, we compute the product of \((-6)(-5)\): $$ f(-3, -5) = (-6) + (-15) - 90 $$ Finally, we add and subtract the values: $$ f(-3, -5) = -6 - 15 - 90 = -111 $$ Now we have the final value: $$ f(-3, -5) = -111 $$

Key Concepts

Functions

In Discrete Mathematics, a function can be thought of as a special relationship between two sets. In our particular example, the function \(f : \mathbf{Z} \times \mathbf{Z} \rightarrow \mathbf{Z}\) defines a map from pairs of integers to a single integer. Here, the rule that associates these elements is given via the expression \(f(x, y) = 2x + 3y - 6xy\). Functions like this are used to specify how to link each pair of inputs with a single output. Understanding functions involves recognizing how changes in input affect the output, a crucial skill for problem-solving in mathematics.

Integer Operations

Integer operations are basic arithmetic calculations performed on whole numbers. These operations include addition, subtraction, multiplication, and division, though division is not present in our function. In our function:

• Addition: We see this in \(2x + 3y\), where constants are added to multiplied forms of our inputs \(x\) and \(y\).
• Multiplication: Notice how each term involves multiplying either by 2 or 3, and in the product \(-6xy\).

In this exercise, we calculate \(f(-3, -5)\) by performing these operations systematically on the inputs \(x = -3\) and \(y = -5\). Negative numbers add complexity, but careful attention ensures accurate solutions.

Arithmetic Expressions

Arithmetic expressions involve mathematical phrases containing numbers and operation symbols. In our example, the expression \(2x + 3y - 6xy\) combines multiple operations.

• At first, substitute \(x\) and \(y\) with -3 and -5 respectively, resulting in \(f(-3, -5) = 2(-3) + 3(-5) - 6(-3)(-5)\).
• Each operation must be performed respecting the order of operations (also known as BODMAS/BIDMAS), which stands for Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication, Addition and Subtraction.

Understanding arithmetic expressions' structure is essential to properly solve them and avoid errors, especially when negative numbers are involved.

Problem Solving Steps

Solving mathematical problems systematically involves taking logical sequential steps. Here's how we approached the problem of finding \(f(-3, -5)\):

• Step 1: Identify the function and its inputs. Locate \(f(x, y) = 2x + 3y - 6xy\) and substitute \(x = -3\) and \(y = -5\).
• Step 2: Simplify expressions step by step. Start by performing the multiplications, then the additions and subtractions.
• Step 3: Calculate the final result of all arithmetic operations, which for this function gives \(f(-3, -5) = -111\).

Careful, step-by-step solutions help organize thoughts, making even complex problems manageable. This clear structure also aids in ensuring accuracy and helps identify any errors in reasoning or calculation.