Problem 29 - Discrete Mathematics with Applications

Question

Prove by the existence method. There are integers \(x\) such that \(x^{2}=x\)

Step-by-Step Solution

Answer

Using the existence method, we have found two integers that satisfy the equation \(x^2 = x\): \(x = 0\) and \(x = 1\). By finding these examples, we have proven that there exist integers \(x\) such that their square is equal to themselves.

Step 1: Understand the equation

We are given the equation \(x^2 = x\). We need to find integers that satisfy this equation.

Step 2: Consider a case where x = 0

Let's see if the integer \(x = 0\) satisfies the equation. Plugging it in, we have: \(0^2 = 0\) This equation is true, so the integer \(x = 0\) satisfies the given equation.

Step 3: Consider a case where x = 1

Now, let's see if the integer \(x = 1\) satisfies the equation. Plugging it in, we have: \(1^2 = 1\) This equation is also true, so the integer \(x = 1\) satisfies the given equation.

Step 4: Conclude the proof using the existence method

We have found two integers, \(x = 0\) and \(x = 1\), that satisfy the equation \(x^2 = x\). By finding these examples, we have proven that there exist integers \(x\) such that their square is equal to themselves, \(x^2 = x\). This completes the proof using the existence method.

Key Concepts

Proof by Example

The concept of 'proof by example' is a method used in mathematics to demonstrate the veracity of a statement by providing specific instances in which the statement holds true. In the case of proving statements about numbers, such as finding integer solutions to an equation, providing even a single example can sometimes be enough to prove the existence of such a solution. However, it is important to note that this type of proof is sufficient only when the goal is to show the existence of something, rather than proving that a condition applies to all possible cases.

For instance, in our exercise, we aim to show that there exists at least one integer value for which the equation \(x^2 = x\) is true. By demonstrating this with the integers 0 and 1, we have used 'proof by example' to confirm the statement's validity. This method aligns perfectly with the existence method proof, as it requires we only need to find one or more examples to substantiate the claim.

Integer Solutions

Finding integer solutions to an equation means determining the whole number or integers that satisfy the given equation. In basic algebra, equations can often be designed or rearranged to find integer solutions. For the equation \(x^2 = x\), we are searching for values of \(x\) that, when squared, yield the original number.

Integer solutions are distinct because they exclude fractions, decimals, and irrational numbers. As showcased in the solution steps, both \(x = 0\) and \(x = 1\) are integer solutions to the equation as they satisfy the criteria without resulting in non-integer values. When exploring integer solutions, it's also essential to consider the context; while some equations have a finite number of integer solutions, others might possess infinitely many.

Basic Algebra

Basic algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols; it is the foundation of more advanced topics in mathematics. Algebraic equations can vary in complexity, from simple linear equations to more complex polynomials. The equation at hand, \(x^2 = x\), is a straightforward algebraic equation where the task is to solve for \(x\).

To solve this equation, we rely on basic algebra principles such as the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero. Simplifying the given equation leads to \(x(x - 1) = 0\), which by the zero-product property, gives us the integer solutions \(x = 0\) or \(x = 1\). This process showcases the power of basic algebra in finding solutions to what might at first seem like a complex problem.