Question

If \(\mathrm{a}=\\{1,2,3\\}\) and \(\mathrm{b}=\\{5,6\\}\), find \(\mathrm{a} \times \mathrm{b}\) and \(\mathrm{b} \times \mathrm{a}\).

Short answer

a \(\times\) b = \(\{(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6)\}\) b \(\times\) a = \(\{(5, 1), (5, 2), (5, 3), (6, 1), (6, 2), (6, 3)\}\)

Step-by-step solution

Determine the Elements in the Cartesian Product (a x b)

To find the Cartesian product a x b, we will form ordered pairs (x, y) such that x ∈ a and y ∈ b. That is, we will create pairs with the first element from set a and the second element from set b.

Generate the Ordered Pairs for a x b

Using the elements in sets a and b, we can create the following ordered pairs for a x b: (1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6)

Determine the Elements in the Cartesian Product (b x a)

Now, we will find the Cartesian product b x a by forming ordered pairs (x, y) such that x ∈ b and y ∈ a. This time, we will create pairs with the first element from set b and the second element from set a.

Generate the Ordered Pairs for b x a

Using the elements in sets b and a, we can create the following ordered pairs for b x a: (5, 1), (5, 2), (5, 3), (6, 1), (6, 2), (6, 3)

Write the Final Results

Finally, we can write the Cartesian products for a x b and b x a as follows: a × b = {(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6)} b × a = {(5, 1), (5, 2), (5, 3), (6, 1), (6, 2), (6, 3)}

Key concepts

Ordered Pairs

In set theory, an ordered pair is a two-element tuple where the order of elements matters. This means that the ordered pair \((x, y)\) is not the same as \((y, x)\) unless \(x = y\). Ordered pairs are fundamental in defining Cartesian products, where each pair consists of one element from each of the given sets.
For example:

• In our case, set \(a\) includes {1, 2, 3}, and set \(b\) includes {5, 6}.
• The ordered pairs \((1, 5)\) or \((5, 1)\) take into account the first element and the second element separately, showing their dependency on the order.

The concept of order allows us to understand different relationships between such pairs, which play a crucial role in various mathematical applications, including functions and relations.

Set Theory

Set theory is a branch of mathematical logic that studies sets, collections of objects considered as objects in their own right. A set can be anything from numbers, symbols, or even other sets. It forms the foundational language for mathematics.
In our problem:

• We have two sets, \(a = \{1, 2, 3\}\) and \(b = \{5, 6\}\).
• The focus is on forming the Cartesian product, which involves creating ordered pairs between members of these two sets.

The operation of generating a Cartesian product itself highlights a keystone idea of set theory — understanding the interaction and connections between different sets through their elements. This is crucial for advancing further into topics such as topology, model theory, and other advanced mathematical concepts.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are distinct and separable. Unlike continuous mathematics, which deals with objects that can vary smoothly, discrete mathematics concerns countable, distinct, and often finite structures.
The Cartesian product, as tackled in the exercise involving sets \(a\) and \(b\), is a prime example of discrete mathematics in action.

• We chose elements individually from sets \(a\) and \(b\) to form discrete units called ordered pairs.
• These pairs are countable and finite — emphasizing discrete math principles.

Such discrete structures help in studying computer science topics, where finite steps and distinct groupings are required, such as algorithms or network theory. Understanding these forms of mathematics can lay a strong theoretical foundation for computer science, cryptography, and even complex logical theories.