Question

Given \(\mathrm{A}=\\{4,9\\}\) and \(\mathrm{B}=\\{1,2\\}\), find all the subsets of \(\mathrm{A} \times \mathrm{B}\)

Short answer

The subsets of \(A \times B = \{(4,1), (4,2), (9,1), (9,2)\}\) are: \[\{\emptyset, \{(4,1)\}, \{(4,2)\}, \{(9,1)\}, \{(9,2)\}, \{(4,1),(4,2)\}, \{(4,1),(9,1)\}, \{(4,1),(9,2)\}, \{(4,2),(9,1)\}, \{(4,2),(9,2)\}, \{(9,1),(9,2)\}, \{(4,1),(4,2),(9,1)\}, \{(4,1),(4,2),(9,2)\}, \{(4,1),(9,1),(9,2)\}, \{(4,2),(9,1),(9,2)\}, \{(4,1),(4,2),(9,1),(9,2)\}\}\]

Step-by-step solution

Define Cartesian Product:

The cartesian product is a mathematical operation on two sets, where the result is a set containing all possible ordered pairs where the first element comes from the first set and the second element comes from the second set. In this case, the cartesian product is: \(\mathrm{A} \times \mathrm{B}\).

Calculate the Cartesian Product:

To calculate the cartesian product, pair each element in set \(\mathrm{A}\) with each element in set \(\mathrm{B}\). In this case, \(\mathrm{A}=\{4, 9\}\) and \(\mathrm{B}=\{1, 2\}\). Thus, the cartesian product, \(\mathrm{A} \times \mathrm{B}\), is: $$\{(4, 1), (4, 2), (9, 1), (9, 2)\}$$

Identify All Subsets:

Now, we want to find all possible subsets of \(\mathrm{A} \times \mathrm{B}\), which means finding all combinations of elements in our cartesian product. We start with an empty subset, then we build subsets with one element, two elements, and so on, until we reach the maximum number of elements, which is equal to the size of \(\mathrm{A} \times \mathrm{B}\). In this case, since \(\mathrm{A} \times \mathrm{B}\) contains 4 elements, we have: - Empty subset: \(\emptyset\) - Subsets with 1 pair: \(\{(4, 1)\}, \{(4, 2)\}, \{(9, 1)\}, \{(9, 2)\}\) - Subsets with 2 pairs: \(\{(4, 1), (4, 2)\}, \{(4, 1), (9, 1)\}, \{(4, 1), (9, 2)\}, \{(4, 2), (9, 1)\}, \{(4, 2), (9, 2)\}, \{(9, 1), (9, 2)\}\) - Subsets with 3 pairs: \(\{(4, 1), (4, 2), (9, 1)\},\{(4, 1), (4, 2), (9, 2)\},\{(4, 1), (9, 1), (9, 2)\},\{(4, 2), (9, 1), (9, 2)\}\) - Subsets with 4 pairs: \(\{(4, 1), (4, 2), (9, 1), (9, 2)\}\) So all subsets of \(\mathrm{A} \times \mathrm{B}\) are as follows: $$\{\emptyset, \{(4, 1)\}, \{(4, 2)\}, \{(9, 1)\}, \{(9, 2)\}, \{(4, 1), (4, 2)\}, \{(4, 1), (9, 1)\}, \{(4, 1), (9, 2)\}, \{(4, 2), (9, 1)\}, \{(4, 2), (9, 2)\}, \{(9, 1), (9, 2)\}, \{(4, 1), (4, 2), (9, 1)\}, \{(4, 1), (4, 2), (9, 2)\}, \{(4, 1), (9, 1), (9, 2)\}, \{(4, 2), (9, 1), (9, 2)\}, \{(4, 1), (4, 2), (9, 1), (9, 2)\}\}$$

Key concepts

Ordered Pairs

An ordered pair is a fundamental concept in mathematics that consists of two elements with a particular sequence. The order in which the elements appear is crucial, such that the ordered pair \(a, b\) is not the same as \(b, a\) unless \(a\) is equal to \(b\). Ordered pairs are often used to represent coordinates on a graph, solutions to equations, and relationships between elements in set theory.

In the cartesian product of two sets, the elements of each set are paired together to form ordered pairs. If we take sets \(A\) and \(B\) from our exercise, the cartesian product \(A \times B\) results in the set of all ordered pairs where the first element is from \(A\) and the second is from \(B\). It's essential to pay attention to the sequence of elements to ensure that all unique combinations are accounted for in the resulting set.

Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. In set theory, the concept of a set is considered to be fundamental, and its properties are defined axiomatically. The language of set theory is used throughout mathematics to express ideas and properties in a precise manner.

The exercise illustrates a fundamental operation in set theory, the cartesian product. The sets involved are \(A\) and \(B\), and we find the cartesian product \(A \times B\) by taking each element of \(A\) and pairing it with each element in \(B\), which results in a new set comprising ordered pairs, as discussed in the previous section. This new set of pairs can then be examined further, such as finding its subsets, which is the crux of this exercise.

Subsets

Subsets are parts of a set that contain some or none of the elements of the parent set. A set \(X\) is considered a subset of set \(Y\) if every element of \(X\) is also an element of \(Y\). Notably, every set is a subset of itself, and the empty set (denoted by \(\emptyset\)) is a subset of every set.

In the context of our exercise, finding all subsets of the cartesian product \(A \times B\) involves identifying all possible combinations of its elements. This includes the empty set, single-element subsets, and so on, up to the subset that contains all elements (the entire set itself). Recognizing and enumerating these subsets are important for comprehending the structure and relationships between the elements within the set.

Mathematical Operations

Mathematical operations are procedures or functions that take one or more input values and produce a single output value. Common operations include addition, subtraction, multiplication, and division. However, in set theory, operations refer to ways of combining, relating, or modifying sets to produce new sets.

The exercise showcases the cartesian product, an operation on two sets that generates a new set of ordered pairs. It is a foundation for more advanced concepts in mathematics, such as relations and functions. Understanding how to perform the cartesian product and identify subsets thereof is a key skill in various mathematical fields, including algebra, combinatorics, and discrete mathematics.