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Chapter 12: Problem 95

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Graph: \(f(x)=\frac{2 x^{2}-x-1}{x^{2}+2 x+1}\)

Short Answer

Expert verified
\( A \cap B = \{ 6, 12, 18, 24, 30 \} \)

Step by step solution

01

- Understand Set A

Set A contains the even numbers between 2 and 30. We can write it as follows: \[ A = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30\} \]
02

- Understand Set B

Set B contains the multiples of 3 between 3 and 30. We can write it as follows: \[ B = \{3, 6, 9, 12, 15, 18, 21, 24, 27, 30\} \]
03

- Find Common Elements (Intersection)

To find the intersection of sets A and B (denoted as \( A \cap B \)), identify the common elements in both sets. Compare the elements of sets A and B: Common elements are 6, 12, 18, 24, and 30.
04

- Write the Intersection

Write the common elements found in step 3 as the intersection of sets A and B. Thus, \[ A \cap B = \{ 6, 12, 18, 24, 30\} \]

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Key Concepts

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

Sets in Mathematics
In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 can be considered as elements of a set. Sets are usually denoted by curly braces \(\{ \}\) and the elements inside are listed, separated by commas. For instance, the set containing 1, 2, and 3 is written as \(\{1, 2, 3\}\).

Sets can represent a variety of things. Some sets are finite, containing a specific number of elements. Others are infinite, with an infinite number of elements. Set theory is fundamental to mathematics because it forms the building blocks for more complex concepts.

When working with sets, we may perform operations such as union (\(\cup\)), intersection (\(\cap\)), and difference (\( - \)).
  • Union (\(\cup\)): Combines all elements from both sets.
  • Intersection (\(\cap\)): Contains only elements common to both sets.
  • Difference (\( - \)): Elements in one set but not in the other.
Let's move on to a practical example involving sets.
Even Numbers
An even number is an integer that is exactly divisible by 2. In other words, when an even number is divided by 2, there is no remainder. Examples of even numbers are 2, 4, 6, 8, and so on.

In set notation, if we were to list the even numbers between 2 and 30, it would look like this: \(\{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30\}\). Notice that each number in this set follows the rule of divisibility by 2.

Understanding even numbers is crucial for comprehending various mathematical concepts and problems, including set operations. These numbers come into play in different types of numerical sequences and patterns we may encounter.
Multiples of Numbers
Multiples of a number are the results of multiplying that number by integers. For example, the multiples of 3 include 3, 6, 9, 12, and so on. To get these multiples, you simply multiply 3 by 1, 2, 3, etc. Mathematically, a multiple of 3 can be represented as \(3n\), where \(n\) is any integer.

If we want to list the multiples of 3 between 3 and 30, we write: \(\{3, 6, 9, 12, 15, 18, 21, 24, 27, 30\}\). These numbers are all divisible by 3 without leaving a remainder.

Understanding multiples is fundamental in various arithmetic operations and can help in finding common factors. When we look at intersections of sets, knowing the multiples can help identify shared elements quickly. In our exercise example, creating a set of multiples and another set of even numbers allows us to find elements that appear in both sets.

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Most popular questions from this chapter

Financial Planning A recently retired couple needs \(\$ 12,000\) per year to supplement their Social Security. They have \(\$ 300,000\) to invest to obtain this income. They have decided on two investment options: AA bonds yielding \(5 \%\) per annum and a Bank Certificate yielding \(2.5 \%\). (a) How much should be invested in each to realize exactly \(\$ 12,000 ?\) (b) If, after 2 years, the couple requires \(\$ 14,000\) per year in income, how should they reallocate their investment to achieve the new amount?

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{l} \frac{4}{x}-\frac{3}{y}=0 \\ \frac{6}{x}+\frac{3}{2 y}=2 \end{array}\right. $$

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} 2 x-3 y=-1 \\ 10 x+y=11 \end{array}\right. $$

Mixing Nuts A store sells cashews for \(\$ 5.00\) per pound and peanuts for \(\$ 1.50\) per pound. The manager decides to mix 30 pounds of peanuts with some cashews and sell the mixture for \(\$ 3.00\) per pound. How many pounds of cashews should be mixed with the peanuts so that the mixture will produce the same revenue as selling the nuts separately?

A doctor's prescription calls for the creation of pills that contain 12 units of vitamin \(\mathrm{B}_{12}\) and 12 units of vitamin E. Your pharmacy stocks three powders that can be used to make these pills: one contains \(20 \%\) vitamin \(\mathrm{B}_{12}\) and \(30 \%\) vitamin \(\mathrm{E} ;\) a second, \(40 \%\) vitamin \(\mathrm{B}_{12}\) and \(20 \%\) vitamin \(\mathrm{E}\) and a third, \(30 \%\) vitamin \(\mathrm{B}_{12}\) and \(40 \%\) vitamin \(\mathrm{E}\). Create \(\mathrm{a}\) table showing the possible combinations of these powders that could be mixed in each pill. Hint: 10 units of the first powder contains \(10 \cdot 0.2=2\) units of vitamin \(\mathrm{B}_{12}\).
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