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Chapter 2: Problem 19

Suppose that \(A, B,\) and \(C\) are sets such that \(A \subseteq B\) and \(B \subset C\) . Show that \(A \subset C\) .

Short Answer

Expert verified
\( A \subset C \) since every element of A is also in B and then in C.

Step by step solution

01

- Understand Given Information

You are given that set A is a subset of set B, which is mathematically represented as: \[ A \subseteq B \]. You are also given that set B is a subset of set C, expressed as: \[ B \subset C \].
02

- Define Subset Relationship

Recall the definition of a subset. If a set X is a subset of a set Y, then every element of X is also an element of Y. Symbolically, for all elements x: \[ x \in X \Rightarrow x \in Y \].
03

- Combine Given Relationships

Since \(A \subseteq B\), every element in A is in B. Similarly, since \(B \subset C\), every element in B is in C. Therefore, any element that is in A must also be in B, and in turn in C. This can be written as: \[ x \in A \Rightarrow x \in B \Rightarrow x \in C \].
04

- Conclude the Subset Relation

From the above, it follows that every element of A is also an element of C. Consequently, A is a subset of C, represented as: \[ A \subset C \].

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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 branch of mathematical logic that studies sets, which are collections of objects. A set is defined by its elements, and these elements can be anything: numbers, people, or even other sets. Understanding subsets is a key concept in set theory. A set \(A\) is called a subset of a set \(B\) if all elements of \(A\) are also elements of \(B\). This is written as \(A \subseteq B\). Set theory allows us to understand relationships between different groups of objects and has many applications in fields such as computer science, statistics, and algebra.

For example, the sets \(A\) and \(B\) have a subset relationship if:
  • Every element in \(A\) is also in \(B\)

    This means if an element belongs to \(A\), it must automatically belong to \(B\), ensuring the 'subset' condition is satisfied. Such relationships form the basis of more complex set operations like union, intersection, and difference.
Proof Techniques
Proof techniques are methods used to establish the truth of mathematical statements. One fundamental proof technique used here is direct proof. In direct proof, you demonstrate a statement is true by a logical progression from known facts. The exercise involves proving \(A \subset C\) given \(A \subseteq B\) and \(B \subset C\).

Here, the steps are:
  • Start with the given information: \(A \subseteq B\) and \(B \subset C\)
  • By the definition of subsets, every element in \(A\) is in \(B\), and every element in \(B\) is in \(C\)
  • So, every element of \(A\) is automatically in \(C\)

    Hence, you conclude \(A \subset C\). This step-by-step logical reasoning is a cornerstone of direct proof in mathematics. It helps in clearly showing the necessary intermediate steps leading to the final conclusion.
Transitivity
Transitivity is a property that applies to relations indicating that if relation holds between a first and a second element, and the same relation holds between the second and a third element, then it also holds between the first and the third element. In simpler terms, if \(A \subseteq B\) and \(B \subset C\), then \(A \subset C\).

A practical example: Suppose you have three buckets. You know that bucket A fits within bucket B, and bucket B fits within bucket C. By transitivity, bucket A must fit within bucket C.
  • Start with the set relationship: \(A \subseteq B\)
  • Combine it with: \(B \subset C\)
  • Use the transitivity property to conclude: \(A \subset C\)

    Transitivity makes it easy to understand how complex relationships can be broken down into simpler, component relations.

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Most popular questions from this chapter

Find \(\bigcup_{i=1}^{\infty} A_{i}\) and \(\bigcap_{i=1}^{\infty} A_{i}\) if for every positive integer \(i\) a) \(A_{i}=\\{-i,-i+1, \ldots,-1,0,1, \ldots, i-1, i\\}\) b) \(A_{i}=\\{-i, i\\}\) c) \(A_{i}^{\prime}=[-i, i],\) that is, the set of real numbers \(x\) with \(-i \leq x \leq i .\) d) \(A_{i}=[i, \infty),\) that is, the set of real numbers \(x\) with \(x \geq i .\)

Show that if \(A\) and \(B\) are finite sets, then \(A \cup B\) is a finite set.

Draw the graph of the function \(f(x)=\lceil x\rceil+\lfloor x / 2\rfloor\) from \(\mathbf{R}\) to \(\mathbf{R} .\)

How many bytes are required to encode \(n\) bits of data where \(n\) equals $$\begin{array}{llll}{\text { a) } 7 ?} & {\text { b) } 17 ?} & {\text { c) } 1001 ?} & {\text { d) } 28,800 ?}\end{array}$$

Data are transmitted over a particular Ethernet network in blocks of 1500 octets (blocks of 8 bits). How many blocks are required to transmit the following amounts of data over this Ethernet network? (Note that a byte is a synonym for an octet, a kilobyte is 1000 bytes, and a megabyte is \(1,000,000\) bytes.) a) 150 kilobytes of data b) 384 kilobytes of data c) 1.544 megabytes of data d) 45.3 megabytes of data
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