Problem 16 - Discrete Mathematics with Applications

Question

Prove each using the law of the contrapositive. If the square of an integer is odd, then the integer is odd.

Step-by-Step Solution

Answer

The given statement is "If the square of an integer is odd, then the integer is odd". The contrapositive is "If the integer is not odd (even), then the square of the integer is not odd (even)." Let n be an even integer, so n = 2k, where k is an integer. Then, \(n^2\) = (2k)^2 = 4(k^2), which is even. Thus, the contrapositive is true, implying that the original statement is also true.

Step 1: Identify the contrapositive statement

The given statement is "If the square of an integer is odd, then the integer is odd". According to the laws of contrapositive, "If not Q, then not P," which in this case is "If the integer is not odd, then the square of the integer is not odd."

Step 2: Choose an even integer

Suppose n is an even integer. By definition, this means that n can be written in the form n = 2k, where k is an integer.

Step 3: Square the even integer

Now, we will calculate the square of n (n^2). Since n is even (n=2k), we can substitute that value in the equation: \(n^2\) = (2k)^2

Step 4: Simplify the equation

Simplify the equation obtained in step 3: \(n^2\) = (2k)(2k) = 4(k^2)

Step 5: Analyze the result

The result we obtained, \(n^2\) = 4(k^2), shows that the square of n is a multiple of 4, which means it must be even (since 4 is even). Therefore, we have proved that "If the integer is not odd, then the square of the integer is not odd." By the law of contrapositive, this implies that the original statement "If the square of an integer is odd, then the integer is odd" is also true.

Key Concepts

Proof by Contrapositive

Understanding the proof by contrapositive involves grasping the logical equivalence between a statement and its contrapositive. In mathematical logic, for a conditional statement 'If P, then Q' the contrapositive is 'If not Q, then not P'. Both statements are logically equivalent, which means that proving the contrapositive is as valid as proving the original statement.

In practice, proving the contrapositive can sometimes be easier because it might involve dealing with simpler or more familiar concepts. It's a powerful tool in a mathematician's toolbox, especially when direct proof is difficult to execute. The ability to think in terms of contrapositive statements also strengthens overall problem-solving skills in mathematics.

Even and Odd Integers

The distinction between even and odd integers is fundamental in number theory, and it plays a crucial role in various mathematical proofs. An even integer is any integer that can be divided by 2 without leaving a remainder, often represented as 2k where k is an integer. Conversely, an odd integer cannot be evenly divided by 2 and can be represented as 2k+1.

Understanding the properties of even and odd integers helps in simplifying and solving many problems. For example, the sum of two even integers is always even, while the sum of an even integer and an odd integer is always odd. This knowledge is especially crucial when working with algebraic expressions and proofs, as you can often make quick conclusions about the parity of an expression based upon its terms.

Mathematical Proof

A mathematical proof is a deductive argument for a mathematical statement. Proofs are the very essence of mathematics and involve a step-by-step logical reasoning process that establishes the truth of the assertion being made. They can be direct, using definitions and theorems to prove something directly, or indirect, such as proof by contrapositive or proof by contradiction.

Proofs ensure that mathematical statements are irrefutable and universally acceptable. When learning or performing proofs, it's important to follow a logical flow and to express each step clearly, since the goal is to convince the reader of the statement's validity based on previously established facts and logical operations. Comprehensive knowledge and skill in crafting mathematical proofs is what allows mathematicians to build on existing theory and introduce new concepts.

Square of an Integer

The square of an integer can be thought of as the number multiplied by itself. It is a fundamental concept in mathematics and plays a crucial role in various branches such as algebra, geometry, and number theory. When you square an even integer, the result is always even, and when you square an odd integer, the result is always odd—which is directly tied to the properties of even and odd numbers.

For instance, squaring an even integer like 2 results in 4, and squaring an odd integer like 3 results in 9. This property becomes very useful in proofs, as seen with the exercise provided where squaring an even number resulted in an even number, thus reinforcing the initial claim through the proof by contrapositive.