Question

Without changing their meanings, convert each of the following sentences into a sentence having the form "If \(P\), then \(Q .\) " An integer is divisible by 8 only if it is divisible by 4

Short answer

The sentence 'An integer is divisible by 8 only if it is divisible by 4' can be converted into the form 'If P, then Q' as: 'If an integer is divisible by 4, then it is divisible by 8'.

Step-by-step solution

Identify P and Q

The given sentence is: 'An integer is divisible by 8 only if it is divisible by 4'. Here, the main conditional part is 'An integer is divisible by 8', which depends on the condition 'it is divisible by 4'. So, we can take 'An integer is divisible by 4' as P and 'It is divisible by 8' as Q.

Formulate the sentence in 'If P, then Q' format

Now that we have identified P and Q, let's formulate the sentence in the 'If P, then Q' format. The new sentence will be: 'If an integer is divisible by 4 (P), then it is divisible by 8 (Q)'.

Key concepts

Logic in Mathematics

At the very heart of mathematics lies logic, a foundational pillar that governs the way mathematical statements and proofs are structured. Logic is about reasoning and the way we connect ideas to arrive at valid conclusions.

One of the most fundamental logical formulations is the conditional statement, often expressed in the form 'If P, then Q.' In this construction, P represents a hypothesis or antecedent, and Q represents a conclusion or consequent. The idea is that the truth of Q depends on the truth of P. If an integer being divisible by 4 (P) is true, then according to our conditional statement, the integer being divisible by 8 (Q) must also be true.

Understanding the conditional relationship is crucial in mathematical reasoning because it dictates the flow of arguments. It's essential to note that if P is true and the statement 'If P, then Q' is true, then Q must be true. However, if P is false, we cannot derive any definitive conclusion about Q. It's a unidirectional link, not a bidirectional one.

Proof Writing

Proof writing is the art of formulating convincing arguments that demonstrate the truth or falsity of a mathematical statement. Good proof relies not only on the logical sequence of propositions but also on clarity and simplicity so that the reader can follow and understand the argument.

When engaging in proof writing, one often uses conditional statements to establish these logical sequences. For instance, the statement 'If an integer is divisible by 4, then it is divisible by 8' can be a step in a proof about divisibility rules.

In proving such a statement, you would start by assuming the 'if' part, known in mathematics as the hypothesis. Next, through logical deductions and perhaps employing known theorems, you would strive to prove the 'then' part, the conclusion. If successful, it forms a logically sound link that can then be used as a building block in a broader proof.

Divisibility Rules

Divisibility rules are handy shortcuts that help determine whether a given integer can be divided by another number without a remainder. These rules are based on the properties of numbers and play a critical role in simplifying complex arithmetic and algebraic processes.

For example, a basic rule states that an integer is divisible by 4 if its last two digits form a number that is divisible by 4. Extending this, a rule for divisibility by 8 requires that the last three digits of an integer form a number divisible by 8.

This connects to our conditional statement: if an integer satisfies the condition of divisibility by 4 (P), it doesn't necessarily satisfy divisibility by 8 (Q). However, if we know that 'If an integer is divisible by 4, then it is divisible by 8' is a true statement, then proving P automatically gives us Q. Recognizing such relationships between divisibility rules accelerates mental calculations and problem-solving in mathematics.