Question

Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) Suppose \(a \in Z\). If \(a^{2}\) is not divisible by \(4,\) then \(a\) is odd.

Short answer

Having started with the assumption that \(a\) is not odd (i.e., \(a\) is even), you can express \(a\) as \(2k\). Squaring this expression results in \(a^{2} = 4k^{2}\), which shows that \(a^{2}\) is divisible by 4.

Step-by-step solution

Identify the contrapositive

Identify the contrapositive of the statement 'If \(a^{2}\) is not divisible by 4, then \(a\) is odd.' The contrapositive is 'If \(a\) is not odd, then \(a^{2}\) is divisible by 4.'

Structure the proof

Proofs generally start by stating what you assume, and then logically deducing the required result. In this case, you start by assuming that \(a\) is not odd, which means \(a\) is even.

Use the property of an even number

Use the property that every even number can be expressed as \(2k\), where \(k\) is an integer. Hence \(a\) can be written as \(2k\).

Determine \(a^{2}\)

Find the square of \(a\), i.e., \(a^{2} = (2k)^{2}\). Simplifying this gives \(a^{2} = 4k^{2}\).

Prove the contrapositive

The resulting expression \(a^{2} = 4k^{2}\) proves that \(a^{2}\) is divisible by 4. This completes the contrapositive statement that if \(a\) is not odd, then \(a^{2}\) is divisible by 4, hence proving the original statement by contrapositive proof.

Key concepts

Mathematical Proof

Understanding mathematical proofs is essential for diving into the deep end of mathematical reasoning. Specifically, contrapositive proof is one of the several types of proofs used to establish the validity of mathematical statements. A mathematical proof is a logical argument that begins with assumptions, known as premises, and uses the rules of logic to arrive at a conclusion.

In the world of mathematics, nothing is assumed to be true unless it can be demonstrated conclusively. Proofs serve as the backbone of mathematical logic, providing concrete evidence and rational certainty.

A proof by contrapositive starts with the contrapositive of the given statement and then demonstrates that this contrapositive is true. If the contrapositive is true, the original statement must be true as well. This logical flip—a staple in mathematical proofs—ensures that our logical building blocks are solid and reliable.

Divisibility

Divisibility is a fundamental concept in number theory, which deals with the properties of integers. It's about understanding when one integer can be divided by another without leaving a remainder. When a number a divides another number b exactly, we say that b is divisible by a.

This concept is crucial for many areas of mathematics, including prime numbers, greatest common divisors, and factorization. For instance, in our exercise, determining whether the square of an integer is divisible by 4 is a direct application of divisibility. The notation often used is a | b, which reads 'a divides b'.

Essential Properties of Divisibility

• If a | b, and b | c, then a | c
• If a | b, then a divides any multiple of b
• a | b and a | c implies that a divides any linear combination of b and c

Even and Odd Numbers

The distinction between even and odd numbers is one of the most elementary, yet vital, concepts in arithmetic. Even numbers are integers that are exactly divisible by 2, like 2, 4, 6, and so forth. In contrast, odd numbers are integers that are not divisible by 2, such as 1, 3, 5, etc.

An important property to remember is that the square of an even number is also even, and the square of an odd number is odd. This property plays a significant role in our exercise on contrapositive proof.

Characteristics of Even and Odd Numbers

• The sum of two even numbers or two odd numbers is always even.
• The sum of an even number and an odd number is always odd.
• The product of two even numbers, two odd numbers, or an even number and an odd number is always even.

Logic in Mathematics

Logic forms the foundation of all mathematical thinking and reasoning. It's the framework that mathematicians use to derive conclusions from a set of given information. There are several logical structures in mathematics, such as direct proofs, indirect proofs (including proofs by contradiction and contrapositive proofs), and proofs by induction.

In our contrapositive proof exercise, logic dictates that for a statement 'if P, then Q', the contrapositive is 'if not Q, then not P'. If either the original statement or its contrapositive is true, the other is true as well. This duality is a key aspect of logic in mathematics and is heavily relied upon in proving statements.

Mastering logical reasoning not only aids in mathematical proofs but also enhances problem-solving skills across various disciplines. It equips students with the tools to discern truths from falsehoods, not just in numbers and equations, but in everyday situations.