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Chapter 2: Problem 31

In Exercises 15-32, express each set using the roster method. \(\\{x \mid x+5=7\\}\)

Short Answer

Expert verified
The set is \(\{2\}\).

Step by step solution

01

Solve the Equation

The given set is described by the mathematical condition \(x + 5 = 7\). Solving for x, subtract 5 from both sides of the equation: \(x + 5 - 5 = 7 - 5\). This simplifies to \(x = 2\)
02

Express the Solution as a Set

Now, according to the roster method, express the solution as a set of the values that satisfy the equation, listed within curly brackets. Here, \(x = 2\) is the only solution, so the set is \(\{2\}\)

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Key Concepts

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

Solving Equations
Solving equations is much like unraveling a mystery. You're presented with a mathematical puzzle and your task is to find the value of the unknown that makes the equation true. Take, for example, the equation in our exercise, where we aim to discover the value of 'x' that satisfies the equation x + 5 = 7.

It's akin to a balance scale; whatever you do to one side, you must do to the other to keep things balanced. So, when we subtract 5 from both sides (x + 5 - 5 = 7 - 5), it maintains that equilibrium and brings us closer to our answer: x = 2. It's essential to carefully perform each mathematical operation on both sides to isolate 'x' and solve the equation. This fundamental process of adding, subtracting, multiplying, or dividing both sides of the equation is the cornerstone of solving equations in algebra.
Set Notation
Set notation is a standard way of defining a collection of elements that share a common characteristic, but when described using the roster method, it's like naming guests at a party by listing their names. The roster method lists out every element in the set, each one separated by a comma, and encloses them in curly brackets.

For example, consider a set of even numbers less than 10. Using the roster method, we'd write it as \(\{2, 4, 6, 8\}\). In the case of our exercise, since there is only one guest at the party鈥攖he number 2鈥攐ur set looks like \(\{2\}\). The curly brackets signal that we are dealing with a set, and every value within them is a part of that set.
Mathematical Condition
Think of a mathematical condition as a 'filter' that only allows numbers meeting certain criteria to be part of a set. This condition can be an equation, an inequality, or another form of a mathematical statement that must be satisfied.

For instance, the mathematical condition in our exercise, x + 5 = 7, required that we find the value of 'x' that, when added to 5, equals 7. It's a bit like saying, 'Who here, when given 5 more, will have 7?' And we found that 'x' must be 2 to fulfill this condition. By solving the equation, we filtered out the single value that meets the criteria, and using the filter of the mathematical condition, we captured the element for our set.

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Most popular questions from this chapter

Describe how a Venn diagram can be used to prove that \((A \cup B)^{\prime}\) and \(A^{\prime} \cap B^{\prime}\) are equal sets.

In Exercises 29-32, use the Venn diagram and the given conditions to determine the number of elements in each region, or explain why the conditions are impossible to meet. \begin{aligned} &n(U)=38, n(A)=26, n(B)=21, n(C)=18 \\ &n(A \cap B)=17, n(A \cap C)=11, n(B \cap C)=8 \\ &n(A \cap B \cap C)=7 \end{aligned}

In Exercises 13-24, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\\B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A^{\prime} \cap\left(B \cup C^{\prime}\right)\)

Research useful websites and present a report on infinite sets and their cardinalities. Explain why the sets of whole numbers, integers, and rational numbers each have cardinal number \(\aleph_{0}\). Be sure to define these sets and show the one-toone correspondences between each set and the set of natural numbers. Then explain why the set of real numbers does not have cardinal number \(\aleph_{0}\) by describing how a real number can always be left out in a pairing with the natural numbers. Spice up the more technical aspects of your report with ideas you discovered about infinity that you find particularly intriguing.

Find each of the following sets. \(A \cap \varnothing\)
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