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Chapter 1: Problem 11

Prove each directly. The product of any even integer and any odd integer is even.

Short Answer

Expert verified
To prove the product of any even integer and any odd integer is even, let the even integer be represented as \(2a\) and the odd integer be represented as \(2b + 1\), where a and b are integers. Multiply them together, \((2a)(2b + 1)\), and apply the distributive property, resulting in \(4ab + 2a\). Factor out the common factor of 2 from the expression: \(2(2ab + a)\). Since \(2ab + a\) is an integer, the entire expression is of the form \(2c\), where \(c\) is an integer (\(c = 2ab + a\)). Hence, the product of any even integer and any odd integer is always even.

Step by step solution

01

Define even and odd integers in general form

An even integer can be represented as 2a, where a is any integer. An odd integer can be represented as 2b + 1, where b is any integer.
02

Multiply the even and odd integers

Now, let's multiply the even and the odd integers together: (2a)(2b + 1)
03

Distributive property

Apply the distributive property to multiply the numbers: = 2a * 2b + 2a * 1
04

Simplify the expression

Now, combine the terms: = 4ab + 2a
05

Factoring out the common factor of 2

We can notice that there's a common factor of 2 in both terms, so we can factor it out: = 2(2ab + a)
06

Conclude the proof

Since 2ab + a is an integer (as it is the sum of the product and an integer), the entire expression is of the form 2c, where c is an integer (c = 2ab + a). Therefore, the product of any even integer and any odd integer is always even.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental principle in algebra that allows us to multiply a number by a sum. This can be particularly helpful when dealing with expressions like \((2a)(2b + 1)\).
  • The property states that multiplying a number by a sum is the same as multiplying each addend by the number and then summing the results.
  • In mathematical terms, it can be expressed as: \(a(b + c) = ab + ac\).
In the context of the problem, the distributive property enables the transformation: \(2a(2b + 1)\) to \((2a \cdot 2b) + (2a \cdot 1)\). This property is essential for expanding and simplifying the expression, making it easier to work with integers and prove the original statement.
Integer Multiplication
Integer multiplication is a straightforward arithmetic operation. When multiplying integers, you combine each term as in the problem, where an even integer \(2a\) is multiplied by an odd integer \(2b + 1\).
  • This operation takes the form \(2a \times 2b + 2a \times 1\).
  • Here, the process involves straightforward multiplication of coefficients and constants.
This step sets the stage for applying further algebraic rules, like the distributive property, to simplify the expression.It's crucial to ensure each multiplication step is handled accurately for a correct outcome in more complex problems.
Factorization
Factorization is a process where you express a number or an expression as a product of its factors. This becomes a vital step in simplifying algebraic expressions, especially when proving statements about even and odd integers.
  • In the solution, after applying the distributive property, we have the expression \(4ab + 2a\).
  • Notice both terms share a common factor of 2, allowing us to factor it out: \(2(2ab + a)\).
By recognizing and factoring out common multiples, we simplify expressions and confirm or "see" that the result \(2(2ab + a)\) is indeed an even number,providing a clear path to the solution.
Proof Techniques
Proof techniques are methods used to establish the truth of a mathematical statement. In this exercise, direct proof is used. Let's break it down:
  • A direct proof begins by assuming the initial premise (in this case, that we have an even and an odd integer) and logically derives the conclusion.
  • Each step follows logically from the last, without assuming the conclusion.
In our problem, we start with expressions representing even and odd integers, apply algebraic rules like the distributive property, and come to the conclusion that the result, \(2(2ab + a)\), is even. Direct proofs are valued for their straightforward and logical sequence of steps, making them essential for learning fundamental problem-solving skills in mathematics.

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Most popular questions from this chapter

Every inhabitant on a mysterious planet is either red or green. In addition, each inhabitant is either male or female. Every red man always tells the truth, whereas every green man always lies. The women, on the other hand, are opposite: every green woman tells the truth and every red woman lies. since the natives always disguise their voices, and wear masks and gloves, it is impossible to identify their sex or color. But a clever anthropologist from Mathland met a native who made a statement from which he was able to deduce that the native was a green woman. (R. Smullyan, Discover, 1993? The second native the anthropologist interviewed also made a statement from which he was able to conclude that the native was a man (but not his color). Give a statement that would work. Again, justify your answer.

The logical operators NAND (not and) and NOR (not or) are defined as follows: $$ \begin{aligned} p & \text { NAND } q \equiv \sim(p \wedge q) \\ p & \text { NOR } q \equiv \sim(p \vee q) \end{aligned} $$ Construct a truth table for each proposition. \(p\) NAND \(q\)

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

Verify that each inference rule is a tautology. $$[(p \rightarrow q) \wedge(q \rightarrow r) | \rightarrow(p \rightarrow r)$$

Draw a switching network with each representation. $$\left(\mathbf{A} \wedge \mathbf{B}^{\prime}\right) \vee\left(\mathbf{A}^{\prime} \wedge \mathbf{B}\right)$$
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