Question

Let \(M=\\{1,2\\}\) and \(N=\\{p, q\\}\). Find (a) \(M \times N\), (b) \(N \times M\), and (c) \(\mathrm{M} \times \mathrm{M}\)

Short answer

(a) \(M \times N = \{(1,p), (1,q), (2,p), (2,q)\}\), (b) \(N \times M = \{(p,1), (p,2), (q,1), (q,2)\}\), and (c) \(M \times M = \{(1,1), (1,2), (2,1), (2,2)\}\).

Step-by-step solution

Find M × N

To find M × N, we need to find all the ordered pairs (m, n) such that m ∈ M and n ∈ N. In this case, M = {1, 2} and N = {p, q}. We can create a table to make it easier to find all the pairs. \[\begin{tabular}{c|c|c|} & p & q \\\hline 1 & (1,p) & (1,q)\\\hline 2 & (2,p) & (2,q)\\ \end{tabular}\] So, M × N = {(1,p), (1,q), (2,p), (2,q)}.

Find N × M

Now, we will find N × M, which means we need to find all the ordered pairs (n, m) such that n ∈ N and m ∈ M. The sets are the same, but the order of the elements is different. We will create another table: \[\begin{tabular}{c|c|c|} & 1 & 2 \\\hline p & (p,1) & (p,2)\\\hline q & (q,1) & (q,2)\\ \end{tabular}\] So, N × M = {(p,1), (p,2), (q,1), (q,2)}.

Find M × M

Finally, we need to find M × M, which means we are looking for ordered pairs (m_1, m_2) such that m_1 ∈ M and m_2 ∈ M. We have the same set for both M values, so we can create a table with M as columns and rows: \[\begin{tabular}{c|c|c|} & 1 & 2 \\\hline 1 & (1,1) & (1,2)\\\hline 2 & (2,1) & (2,2)\\ \end{tabular}\] So, M × M = {(1,1), (1,2), (2,1), (2,2)}. In conclusion, we found that (a) M × N = {(1,p), (1,q), (2,p), (2,q)}, (b) N × M = {(p,1), (p,2), (q,1), (q,2)}, and (c) M × M = {(1,1), (1,2), (2,1), (2,2)}.

Key concepts

Ordered Pair

In mathematics, an ordered pair is a fundamental concept used to construct more complex structures. It is a pair of elements arranged in a specific order, where the position of each element matters. For instance, if we have two elements, 'a' and 'b', the ordered pair (a, b) is not the same as (b, a) unless 'a' and 'b' are identical.

The concept of ordered pairs is crucial in coordinate geometry, where points on a plane are represented by pairs of numbers, typically (x, y), indicating their position on the X and Y axes. It's also an essential component of set theory, particularly when forming the Cartesian product between sets.

In the exercise provided, the Cartesian product creates ordered pairs by combining each element of set M with each element of set N, emphasizing that (1, p) and (p, 1) are distinct ordered pairs due to their sequence.

Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. In the context of set theory, the objects could be numbers, characters, points, or even other sets. Sets are typically described by listing their elements between curly brackets, for example, 'A = \(\{1, 2, 3\}\)'.

A fundamental operation in set theory is the Cartesian product, denoted by '×', which combines elements from two sets to create pairs, generating a new set composed of these ordered pairs. This operation is crucial for many areas of mathematics, including finite mathematics, relations, and functions. The Cartesian product gives a concrete way to explore possible combinations, and in the exercise, we see this with the resulting sets from combining M and N in various ways.

Sets are foundational to mathematics and understanding their behavior, operations, and properties is key to grasping more complex mathematical concepts.

Finite Mathematics

Finite mathematics is an area of study focusing on mathematical concepts applicable to non-mathematical fields such as business, economics, social sciences, and life sciences. It includes subjects like algebra, graph theory, mathematical modeling, statistics, and operations research. Finite mathematics often deals with finite sets, meaning sets that have a limited number of elements.

In the exercise, you encounter finite sets M and N. Working with these sets and understanding their Cartesian products are examples of problems one might encounter in finite mathematics. It’s important because it shows how to apply mathematical reasoning to solve real-world problems that may have a finite number of possible outcomes or options. The concepts of set theory, such as the Cartesian product, play a prominent role in finite mathematics by allowing for the enumeration and analysis of distinct and orderly configurations.