Question

Find \(\mathrm{A} \times \mathrm{B}\) and \(\mathrm{B} \times \mathrm{A}\), for \(\mathrm{A}=\\{1,2,3\\}, \mathrm{B}=\\{\mathrm{a}, \mathrm{b}\\}\).

Short answer

A × B = \{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\} and B × A = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}.

Step-by-step solution

Define the Sets A and B

We are given two sets: Set A = {1, 2, 3} Set B = {a, b}

Calculate A × B

To find the Cartesian product of A and B, A × B, we need to create ordered pairs where the first element is taken from set A, and the second element is taken from set B. A × B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}

Calculate B × A

Similarly, we will find the Cartesian product of B and A, B × A, by creating ordered pairs where the first element is taken from set B, and the second element is taken from set A. B × A = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)} The Cartesian products A × B and B × A are: A × B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)} B × A = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

Key concepts

Ordered Pairs

An ordered pair is a fundamental concept in set theory, consisting of two elements arranged in a specific order. The key feature that makes an ordered pair unique is that the order of the elements matters – if you change the order, you have a different ordered pair.

For example, (1, a) and (a, 1) are considered two distinct ordered pairs because the first elements are different. In the given exercise, we use ordered pairs to form the Cartesian product. An ordered pair is written in parentheses with the elements separated by a comma. It's like a tiny list where the first item is always first, and the second item is always second. This pair uniqueness is quite unlike a set where order is immaterial and repetition is not allowed.

Set Theory

Set theory is a branch of mathematical logic that studies collections of objects, known as sets. Sets are one of the most fundamental and versatile concepts in mathematics. They are purely abstract collections defined by their members, without any particular order or repetition. For instance, in the exercise, set A contains numbers \(1, 2, 3\) while set B contains letters \(a, b\).

The operations we can perform on sets include union, intersection, difference, and complement. But relevant to our exercise is the Cartesian product, which combines sets in a way that results in a set of ordered pairs. Each pair consists of one element from each of the original sets, systematically paired so that every possible combination is included. Set theory holds vast importance across various fields including computer science, probability, and statistics.

Finite Sets

A finite set is a set with a limited number of elements. In other words, you can count the elements, and eventually, you'll run out of them. This contrasts with infinite sets, which go on forever and can't be counted to completion. Our sets A and B from the exercise are both finite sets: set A with three elements \(1, 2, 3\) and set B with two elements \(a, b\).

Finite sets are easy to handle when we're doing operations like finding a Cartesian product since we can list all possible ordered pairs. It gets more complicated with infinite sets since we cannot list out all the elements. Understanding the concept of finite sets is crucial when learning the basics of set theory and when beginning to grasp the nature and behavior of numbers and objects within the mathematical realm.