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Chapter 5: Problem 3

Let \(A=\\{x \in \mathbb{Z} \mid x \equiv 7(\bmod 8)\\}\) and \(B=\\{x \in \mathbb{Z} \mid x \equiv 3(\bmod 4)\\}\). (a) List at least five different elements of the set \(A\) and at least five elements of the set \(B\) (b) Is \(A \subseteq B\) ? Justify your conclusion with a proof or a counterexample. (c) Is \(B \subseteq A\) ? Justify your conclusion with a proof or a counterexample.

Short Answer

Expert verified
(a) The first five positive elements of set A are \(7, 15, 23, 31, 39\). The first five positive elements of set B are \(3, 7, 11, 15, 19\). (b) We found a counterexample (23) that belongs to A but not to B. Thus, we can conclude that \(A \nsubseteq B\). (c) We found a counterexample (11) that belongs to B but not to A. Thus, we can conclude that \(B \nsubseteq A\).

Step by step solution

01

(a) List elements of the sets A and B

To list at least five elements of each set, we must analyze their definitions: Set A is defined as \[A = \{x \in \mathbb{Z} \mid x \equiv 7 (\bmod 8)\}.\] This means that the integer x belongs to set A if x is congruent to 7 modulo 8 (that is, the remainder when x is divided by 8 is 7). Set B is defined as \[B = \{x \in \mathbb{Z} \mid x \equiv 3 (\bmod 4)\}.\] Similarly, this means that the integer x belongs to set B if x is congruent to 3 modulo 4 (that is, the remainder when x is divided by 4 is 3). Now we will list at least five elements for each set: The first five positive elements of set A are \[7, 15, 23, 31, 39.\] The first five positive elements of set B are \[3, 7, 11, 15, 19.\]
02

(b) Determine if A is a subset of B

To check whether A is a subset of B (denoted as \(A \subseteq B\)), we must show that every element in A also belongs to B. If we find a counterexample, i.e., an element that belongs to A but not to B, that would mean that A is not a subset of B. Let's take the second element from set A, which is 15. We see that 15 belongs to both A and B. However, if we take the next element in A, which is 23, we find that its remainder when divided by 4 is 3 (23 / 4 = 5 + 3/4). Therefore, 23 belongs to A but not to B. Thus, we have a counterexample, and we can conclude that \(A \nsubseteq B\).
03

(c) Determine if B is a subset of A

To check whether B is a subset of A (denoted as \(B \subseteq A\)), we must show that every element in B also belongs to A. If we find a counterexample, i.e., an element that belongs to B but not to A, that means B is not a subset of A. Let's take the second element from set B, which is 7. We see that 7 belongs to both B and A. However, if we take the next element in B, which is 11, we find that its remainder when divided by 8 is 3 (11 / 8 = 1 + 3/8). Therefore, 11 belongs to B but not to A. Thus, we have a counterexample, and we can conclude that \(B \nsubseteq A\).

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Key Concepts

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

Subsets and Supersets
In set theory, one of the important concepts to understand is the relationship between subsets and supersets. A set \(A\) is considered a subset of another set \(B\) (denoted as \(A \subseteq B\)) if every element in \(A\) is also in \(B\). In this case, \(B\) could be regarded as a superset of \(A\). For example, if you have a set of fruits like \(\{apple, banana\}\) and another set \(\{apple, banana, cherry\}\), the first set is a subset of the second.

To determine subset and superset relationships between sets, we often try to find a counterexample. A counterexample is an element in \(A\) that is not in \(B\). If at least one such element exists, \(A\) is not a subset of \(B\), and vice versa. Proper understanding of subsets and supersets helps in structuring and reasoning different types of number sets, especially in problems like the one given, where we need to check subsets between two integer sets.
Modulo Arithmetic
Modulo arithmetic, often just referred to as 'mod', deals with divisors and remainders. It is denoted by the operation \(x \equiv b \pmod{m}\), meaning that \(x\) divided by \(m\) leaves a remainder \(b\). This arithmetic is fundamental when dealing with number patterns, particularly in set definitions that employ modular conditions.

Consider the set \(A = \{x \in \mathbb{Z} \mid x \equiv 7 \pmod{8}\}\). This notation tells us that the integers in set \(A\) always leave a remainder of 7 when divided by 8. Understanding this concept lets you generate elements of \(A\) very easily, starting from 7 and adding multiples of 8 (i.e., 7, 15, 23, etc.). The concept applies similarly to set \(B\), where integers are congruent to 3 modulo 4.

Modulo arithmetic is a powerful tool in number theory and allows us to simplify problems by focusing only on the remainders, which gives a cyclic nature to the sets of numbers expressed in this form.
Integer Sets
Integer sets, as the name suggests, consist of whole numbers that can be either positive, negative, or zero. They play a crucial role in various mathematical operations and have specific properties that differentiate them from other types of numbers, such as fractions or decimals.

For example, in the given exercise, sets \(A\) and \(B\) are clearly defined as integer sets under specific modular conditions. Set \(A\) includes integers congruent to 7 modulo 8, while set \(B\) includes those congruent to 3 modulo 4.

Integer sets can be finite or infinite. In the exercise's context, since both sets continue indefinitely following their rules, they are infinite. Utilizing integer sets in algebra and other math fields helps in establishing sequences and logical orders which are irreplaceable in many mathematical insights and solutions.

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Most popular questions from this chapter

Let \(\Lambda\) be a nonempty indexing set and let \(\mathcal{A}=\left\\{A_{\alpha} \mid \alpha \in \Lambda\right\\}\) be an indexed family of sets. (a) Prove that if \(B\) is a set such that \(B \subseteq A_{\alpha}\) for every \(\alpha \in \Lambda\), then \(B \subseteq \bigcap_{\alpha \in \Lambda} A_{\alpha}\) (b) Prove that if \(C\) is a set such that \(A_{\alpha} \subseteq C\) for every \(\alpha \in \Lambda,\) then \(\bigcup_{\alpha \in \Lambda} A_{\alpha} \subseteq C\)

For each natural number \(n,\) let \(A_{n}=\\{k \in \mathbb{N} \mid k \geq n\\} .\) Determine if the following statements are true or false. Justify each conclusion. (a) For all \(j, k \in \mathbb{N},\) if \(j \neq k,\) then \(A_{j} \cap A_{k} \neq \emptyset\) (b) \(\bigcap_{k \in \mathbb{N}} A_{k}=\emptyset\)

Let \(A\) and \(B\) be subsets of some universal set \(U\). (a) Prove that \(A-B\) and \(A \cap B\) are disjoint sets. (b) Prove that \(A=(A-B) \cup(A \cap B)\).

We can extend the idea of consecutive integers (See Exercise (10) in Section 3.5) to represent four consecutive integers as \(m, m+1, m+2,\) and \(m+3,\) where \(m\) is an integer. There are other ways to represent four consecutive integers. For example, if \(k \in \mathbb{Z}\), then \(k-1, k, k+1,\) and \(k+2\) are four consecutive integers. (a) Prove that for each \(n \in \mathbb{Z}, n\) is the sum of four consecutive integers if and only if \(n \equiv 2(\bmod 4)\). (b) Use set builder notation or the roster method to specify the set of integers that are the sum of four consecutive integers. (c) Specify the set of all natural numbers that can be written as the sum of four consecutive natural numbers. (d) Prove that for each \(n \in \mathbb{Z}, n\) is the sum of eight consecutive integers if and only if \(n \equiv 4(\bmod 8)\) (e) Use set builder notation or the roster method to specify the set of integers that are the sum of eight consecutive integers. (f) Specify the set of all natural numbers can be written as the sum of eight consecutive natural numbers.

For each statement, write a brief, clear explanation of why the statement is true or why it is false. (a) The set \(\\{a, b\\}\) is a subset of \(\\{a, c, d, e\\}\). (b) The set \\{-2,0,2\\} is equal to \(\left\\{x \in \mathbb{Z} \mid x\right.\) is even and \(\left.x^{2}<5\right\\}\). (c) The empty set \(\emptyset\) is a subset of \\{1\\} (d) If \(A=\\{a, b\\},\) then the set \(\\{a\\}\) is a subset of \(\mathcal{P}(A)\).
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