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Introduction to the Theory of Computation

Sipser Michael

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Problem 4

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If $A$ has $a$ elements and $B$ has $b$ elements, how many elements are in $A \times B$ ? Explain your answer.

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The number of elements in the Cartesian product $A \times B$ is equal to $a \times b$, where $a$ is the number of elements in A and $b$ is the number of elements in B. This is because each element in A can be paired with each element in B, resulting in a total of $a \times b$ ordered pairs.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Understanding Cartesian Product

A Cartesian product of two sets A and B, denoted as $A \times B$, is the set of all possible ordered pairs where the first element is from A and the second element is from B. The Cartesian product of two sets results in a new set containing all possible combinations of elements from the two sets.

Step 2: Counting Elements in A

Set A has $a$ elements. For each element in A, there will be an ordered pair formed with the elements of B.

Step 3: Counting Elements in B

Set B has $b$ elements. For each element in B, there will be an ordered pair formed with the elements of A.

Step 4: Counting Elements in the Cartesian Product

Since there are $a$ elements in A and $b$ elements in B, there will be $a \times b$ combinations for the ordered pairs. Hence the number of elements in the Cartesian product $A \times B$ is equal to $a \times b$.

Step 5: Result

The Cartesian product $A \times B$ has $a \times b$ elements, where $a$ is the number of elements in A and $b$ is the number of elements in B.

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Chapter 0: Chapter 0

If \(A\) has \(a\) elements and \(B\) has \(b\) elements, how many elements are in $A \times B$ ? Explain your answer.

Short Answer

Step by step solution

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Step 1: Understanding Cartesian Product

Step 2: Counting Elements in A

Step 3: Counting Elements in B

Step 4: Counting Elements in the Cartesian Product

Step 5: Result

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Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

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