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Chapter 0: Problem 33

Determine whether the statement is true or false. a. \(\\{-2,-4,-6\\}\subset\\{-6,-4,-2,0\\}\) b. \(\\{-6,-4,-2,0\\}\subset\\{-2,-4,-6\\}\)

Short Answer

Expert verified
a. True b. False

Step by step solution

01

- Understanding Subsets

A set A is considered a subset of set B if every element of set A is also an element of set B. This is denoted as \(A \subset B\).
02

- Analyzing Part (a)

Check if each element of the set \(\{-2, -4, -6\}\) is also a member of the set \(\{-6, -4, -2, 0\}\).
03

- Verification for Part (a)

Elements of \(\{-2\}\), \(\{-4\}\), and \(\{-6\}\) are indeed in the set \(\{-6, -4, -2, 0\}\). Thus, \(\{-2, -4, -6\}\) is a subset of \(\{-6, -4, -2, 0\}\).
04

- Conclusion for Part (a)

Based on the verification, the statement for part (a) is true.
05

- Analyzing Part (b)

Check if each element of the set \(\{-6, -4, -2, 0\}\) is also a member of the set \(\{-2, -4, -6\}\).
06

- Verification for Part (b)

The element \(0\) in the set \(\{-6, -4, -2, 0\}\) is not in the set \(\{-2, -4, -6\}\). Hence, \(\{-6, -4, -2, 0\}\) is not a subset of \(\{-2, -4, -6\}\).
07

- Conclusion for Part (b)

Based on the verification, the statement for part (b) is false.

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Key Concepts

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

Subset Definition
In Set Theory, a subset is a set where every element is also contained in another set. We use the symbol \(\subset\) to denote that set A is a subset of set B, written as \(A \subset B\). For instance, if we have two sets, A = \(\{-2, -4, -6\}\) and B = \(\{-6, -4, -2, 0\}\), to determine if A is a subset of B, we need to check that every element in A is also in B.

If every element in A is found within B, then A is indeed a subset of B. Conversely, if even a single element in A is not in B, then A is not a subset of B. Understanding this concept is crucial to interpreting and working with sets in mathematics.
Set Membership
Set membership refers to the relationship between an element and a set. If an element a belongs to set A, we write \(a \in A\). Conversely, if an element a does not belong to set A, we write \(a otin A\).

For example, let's use the sets from the exercise: A = \(\{-2, -4, -6\}\) and B = \(\{-6, -4, -2, 0\}\). Here, \(-2 \in A\), \(-4 \in A\), and \(-6 \in A\). Also, since \(A \subset B\), it means every element of A is also in B, hence \(-2 \in B\), \(-4 \in B\), and \(-6 \in B\).

By verifying the membership of each element, we can determine subset relationships accurately.
Verification of Subsets
To verify whether one set is a subset of another, we follow a series of steps:

  • Identify elements of set A.
  • Check if each element of A is also an element of set B.
  • If every element of A is in B, then A is a subset of B, otherwise, it is not.


Looking at part (a) of the exercise: we checked that all elements of A = \(\{-2, -4, -6\}\) are in B = \(\{-6, -4, -2, 0\}\). Since this is true for all elements (i.e., \(-2 \in B\), \(-4 \in B\), \(-6 \in B\)), A is a subset of B.

Now for part (b): we verified if all elements of B = \(\{-6, -4, -2, 0\}\) are also in A = \(\{-2, -4, -6\}\). Here, \(0 otin A\), so B is not a subset of A. This verification process helps in making precise determinations about subset relationships.

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Most popular questions from this chapter

The total national expenditure for health care has been increasing since the year 2000 . For privately insured individuals in the United States, the following models give the total amount spent for health insurance premiums \(I\) (in \$ billions) and the total amount spent on other out-of-pocket health- related expenses \(P\) (in \$ billions). (Source: U.S. Centers for Medicare \& Medicaid Services, www.census.gov) \(I=45.58 x+460.1 \quad\) Total spent on health insurance premiums \(x\) years since 2000. \(P=10.86 x+191.5 \quad\) Other out-of-pocket health-related expenses \(x\) years since 2000. a. Write and simplify the polynomial representing \(I+P\). b. Interpret the meaning of the polynomial from part (a). c. Evaluate the polynomial \(I+P\) for \(x=6,\) and interpret the meaning of this value.

Determine the degree of the polynomial. $$ -8 p^{2} q r^{5}+4 p q^{8} r^{2}+5 p^{3} q^{3} r $$

Determine the degree of the polynomial. $$ -4.7 a b c^{4}-5.2 a^{2} b c^{5}+2.6 a^{3} c $$

Perform the indicated operations and simplify. $$ \left(\frac{1}{6} n-\frac{4}{5} p^{4}\right)\left(\frac{1}{6} n+\frac{4}{5} p^{4}\right) $$

Suppose that \(x\) represents the larger of two consecutive odd integers. a. Write a polynomial that represents the smaller integer. b. Write a polynomial that represents the sum of the two integers. Then simplify. c. Write a polynomial that represents the product of the two integers. Then simplify. d. Write a polynomial that represents the difference of the squares of the two integers. Then simplify.
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