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Chapter 2: Problem 12

Write \(\subseteq\) or \(\nsubseteq\) in each blank so that the resulting statement is true. \(\left\\{\frac{1}{2}, \frac{1}{3}\right\\}\) ___ \(\\{2,3,5\\}\)

Short Answer

Expert verified
The correct symbol to fill in the blank is \(\nsubseteq\). So, \(\left\{\frac{1}{2}, \frac{1}{3}\right\} \nsubseteq \{2,3,5\}\).

Step by step solution

01

Identify Elements

At first, identify the elements of both sets. The first set contains elements \(\frac{1}{2}\) and \(\frac{1}{3}\), while the second set contains elements \(2\), \(3\), and \(5\).
02

Compare Elements

Compare the elements of the first set to the second set. Neither \(\frac{1}{2}\) nor \(\frac{1}{3}\) are elements of the second set.
03

Write the Relation

Since no elements of the first set are found in the second set, it shows that the first set is not a subset of the second set. So, the relation would be noted as: \(\left\{\frac{1}{2}, \frac{1}{3}\right\} \nsubseteq \{2,3,5\}\).

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Key Concepts

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

Set Theory
Set theory is a fundamental part of mathematics that deals with collections of objects, which we call sets. Each object in a set is known as an element. In the exercise, we have two sets to consider:
  • The first set contains \(\frac{1}{2}\) and \(\frac{1}{3}\).
  • The second set comprises the numbers \(2\), \(3\), and \(5\).
Understanding sets involves knowing whether elements of one set belong to another, which brings us to concepts like subsets and superset relationships.
A set is considered a subset of another if every element of the first set is also in the second set. A useful notation in set theory is the subset symbol \(\subseteq\)\, which denotes that one set is a subset of another. Conversely, when none or not all elements are included, we use \(subseteq\)\ to indicate non-subset relationships.
Element Comparison
Element comparison is crucial when determining relationships between sets. This process involves checking if each element of one set is present in another set. Let's consider this step-by-step for better clarity:
  • We start by listing elements of both sets. Here we have the set \(\left\{\frac{1}{2}, \frac{1}{3}\right\}\) and compare these with \(\{2, 3, 5\}\).
  • Each element from the first set is compared against each element in the second set to see if it appears there.
  • In our example, \(\frac{1}{2}\) does not appear alongside \(2, 3, 5\) and neither does \(\frac{1}{3}\).

Because none of the elements from the first set are in the second set, we conclude that the first set is not a subset of the second set. Thus, recognizing the absence of common elements is what helped us decide the final subset relationship.
Mathematical Notation
Mathematical notation provides a universal language to make the communication of concepts straightforward. In our problem, two specific symbols were essential:
  • \(\subseteq\) symbolizes that one set is a subset of another. It means every element of the first set also appears in the second set.
  • \(subseteq\) represents the opposite relationship, signaling that the set on the left side does not have all its elements contained in the set on the right.

These symbols offer a concise way to express complex ideas, replacing detailed descriptions with simple marks. In our solution, recognizing that no element from \(\left\{\frac{1}{2}, \frac{1}{3}\right\}\) overlaps with \(\{2, 3, 5\}\) allows us to succinctly write \(\left\{\frac{1}{2}, \frac{1}{3}\right\} subseteq \{2,3,5\}\), making clear the non-subset relationship.

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Most popular questions from this chapter

Let $$ \begin{aligned} U &=\\{1,2,3,4,5,6,7\\} \\ A &=\\{1,3,5,7\\} \\ B &=\\{1,2,3\\} \\ C &=\\{2,3,4,5,6\\} \end{aligned} $$ Find each of the following sets. \((A \cap B) \cup(A \cap C)\)

In the August 2005 issue of Conswner Reports, readers suffering from depression reported that alternative treatments were less effective than prescription drugs. Suppose that 550 readers felt better taking prescription drugs, 220 felt better through meditation, and 45 felt better taking St. John's wort. Furthermore, 95 felt better using prescription drugs and meditation, 17 felt better using prescription drugs and St. John's wort, 35 felt better using meditation and St. John's wort, 15 improved using all three treatments, and 150 improved using none of these treatments (Hypothetical results are partly based on percentages given in Consumer Reports.) a. How many readers suffering from depression were included in the report? Of those included in the report, b. How many felt better using prescription drugs or meditation? c. How many felt better using St. John's wort only? d. How many improved using prescription drugs and meditation, but not St. John's wort? e. How many improved using prescription drugs or St. John's wort, but not meditation? f. How many improved using exactly two of these treatments? g. How many improved using at least one of these treatments?

In Exercises 41-66, 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. \(A^{\prime} \cap B^{\prime}\)

In Exercises 41-66, 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. \(A \cap B\)

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. \(n(U)=40, n(A)=10, n(B)=11, n(C)=12\) \(n(A \cap B)=6, n(A \cap C)=9, n(B \cap C)=7\) \(n(A \cap B \cap C)=2\)
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