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Chapter 2: Problem 20

Show that if \(|A|=|B|\) and \(|B|=|C|,\) then \(|A|=|C|\)

Short Answer

Expert verified
By the transitive property of equality, \(|A|=|C|\).

Step by step solution

01

- Understand Given Information

Interpret the given information: - \(|A|=|B|\) implies the cardinality (size) of set A is equal to the cardinality of set B.- \(|B|=|C|\) implies the cardinality of set B is equal to the cardinality of set C.
02

- Use Transitive Property of Equality

The transitive property of equality states that if \(a=b\) and \(b=c\), then \(a=c\). Applying this property to the cardinalities:- From \(|A|=|B|\) and \(|B|=|C|\), it follows that \(|A|=|C|\).
03

- Conclude with the Result

Since the transitive property holds true for the cardinalities, we can conclude that \(|A|=|C|\).

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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 fundamental area of mathematics that deals with collections of objects, known as sets. Sets can be anything: numbers, letters, or even other sets. The concept of a set is crucial because it provides a foundation for nearly all of mathematical logic and reasoning.

Remember:
  • A set is typically denoted by curly braces, such as \( A = \{1, 2, 3\} \).
  • The elements of a set are the objects contained in it.

Understanding sets is the first step in exploring more complex mathematical concepts, such as cardinality, which measures the size of a set. If two sets have the same number of elements, they have the same cardinality.
Transitive Property
The transitive property is a critical principle in mathematics. It states that if a first element is related to a second, and the second is related to a third, then the first is related to the third.

Formally, if you have:
  • \( a = b \)
  • and \( b = c \)

Then you can conclude:
  • \( a = c \)

This property is extremely valuable when working with equalities and relationships in mathematics. In our example, it helps us deduce that if the cardinality of set A is equal to set B, and set B is equal to set C, then the cardinality of set A is equal to set C.
Cardinal Number
Cardinal numbers measure the size of sets. Unlike regular numbers, which count specific quantities, cardinal numbers describe the 'how many' aspect of a set.

Here's how to understand cardinal numbers better:
  • Two sets have the same cardinality if you can pair each element of one set with a unique element of the other set, with no elements left unpaired.
  • For instance, set \( A = \{1, 2, 3\} \) and set \( B = \{a, b, c\} \) have the same cardinality because each element in A can be matched to an element in B.

The cardinal number concept is vital in showing the equality of sets. In our exercise, the premise \( |A| = |B| \) and \( |B| = |C| \) allow us to conclude \( |A| = |C| \) using the transitive property. This means the sets A and C have the same number of elements.

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Most popular questions from this chapter

The \(n \times n\) matrix \(\mathbf{A}=\left[a_{i j}\right]\) is called a diagonal matrix if \(a_{i j}=0\) when \(i \neq j .\) Show that the product of two \(n \times n\) diagonal matrices is again a diagonal matrix. Give a simple rule for determining this product.

Show that there is no one-to-one correspondence from the set of positive integers to the power set of the set of positive integers. [Hint: Assume that there is such a one-to-one correspondence. Represent a subset of the set of positive integers as an infinite bit string with ith bit 1 if i belongs to the subset and 0 otherwise. Suppose that you can list these infinite strings in a sequence indexed by the positive integers. Construct a new bit string with its ith bit equal to the complement of the ith bit of the ith string in the list. Show that this new bit string cannot appear in the list.]

Show that matrix addition is commutative; that is, show that if \(\mathbf{A}\) and \(\mathbf{B}\) are both \(m \times n\) matrices, then \(\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}\)

Show that if \(x\) is a real number and \(n\) is an integer, then a) \(x < n\) if and only if \(\lfloor x\rfloor < n .\) b) \(n < x\) if and only if \(n < \lceil x\rceil\)

Show that if \(x\) is a real number and \(m\) is an integer, then \(\lceil x+m\rceil=\lceil x\rceil+ m .\)
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