Problem 8 - Discrete Mathematics with Applications

Question

Mark each sentence as true or false, where \(n\) is an arbitrary nonnegative integer and \(0 \leq r \leq n\). $$(m+n) !=m !+n !$$

Step-by-Step Solution

Answer

The given expression \((m+n)! = m! + n!\) is false for arbitrary nonnegative integers \(m\) and \(n\) under the constraint \(0 \leq r \leq n\), based on the properties of factorials and test cases evaluated.

Step 1: Understanding the given expression

To start, let's analyze the given expression - \((m+n)!=m!+n!\). Here, we have an equation involving the addition of two-factorials \(m!\) and \(n!\), and this sum is being equated to the factorial of their sum \((m+n)!\).

Step 2: Analyzing the properties of factorials

Recall that the factorial function is defined as: \[n! = n \times (n-1) \times (n-2) \times ... \times 1\] Factorial is only defined for nonnegative integers. It has some important properties such as the following: 1. \(0! = 1\). 2. \(1! = 1\). 3. Factorial function is non-linear, i.e., \((a+b)! \neq a!+b!\).

Step 3: Testing the given expression with some values

Let's test the given expression with some nonnegative values of m and n. #### Test case 1: m = 0, n = 0 \[(0+0)! = 0!+0! \\ 1=1+1 \\ 1=2\] The expression is false for this case. #### Test case 2: m = 2, n = 1 \[(2+1)! = 2!+1! \\ 3!=2+1 \\ 6=3\] The expression is false for this case.

Step 4: Conclusion

Based on the properties of factorials and the test cases we evaluated, the given expression \((m+n)!=m!+n!\) is false for arbitrary nonnegative integers m and n under the constraint \(0 \leq r \leq n\).

Key Concepts

Discrete Mathematics

Discrete Mathematics is an area of mathematics that deals with objects that can assume only distinct, separate values. It involves studying mathematical structures that are fundamentally discrete rather than continuous. In other words, it deals with countable, often finite, sets. This field encompasses a wide array of topics such as set theory, logic, graph theory, and combinatorics.

Understanding discrete mathematics is essential for several areas of computer science, cryptography, and even in algorithms that operate within our digital world. Problems in discrete mathematics often involve integers, logical statements, and discrete structures like graphs, where the concept of nonnegative integers and factorial functions also play a pivotal role.

Nonnegative Integers

Nonnegative integers are whole numbers that include zero and all the positive integers. These numbers are crucial in the realm of discrete mathematics because they often represent countable quantities, such as the number of objects in a set or the number of paths between nodes in a graph.

In the context of factorial functions, nonnegative integers serve as the input values. This imposes a critical limitation since factorials for negative numbers are not defined. As a fundamental concept within many mathematical expressions and proofs, understanding nonnegative integers and their behaviors is foundational to grasping bigger concepts in discrete mathematics, including the factorial function.

Factorial Function

The factorial function, denoted by an exclamation mark (!), is a function that multiplies a nonnegative integer by all of the smaller nonnegative integers down to one. The formal definition is:\[n! = n \times (n-1) \times (n-2) \times ... \times 1\]The value of \(0!\) is defined to be \(1\), which is a key convention in mathematical analysis and combinatorics.

Some critical properties of the factorial function include its growth rate, which is more than exponential, and its non-linearity. The common misconception that \((a+b)!\) might somehow be related to \(a!+b!\) can lead to misunderstandings. As explicitly demonstrated in the given exercise, the sum of factorials is not equal to the factorial of the sum, which is a fundamental property to recognize in discrete mathematics.

Understanding the factorial function is vital for calculating permutations and combinations, which quantify the number of ways in which a set or sequence can be ordered or arranged. Thus, it has immense applications in probability, statistics, and various fields that require combinatorial calculations.