If two sets \(\mathrm{A}\) and \(\mathrm{B}\) are having 43 elements in common, then the number of elements common to each of the sets \(\mathrm{A} \times \mathrm{B}\) and \(\mathrm{B} \times \mathrm{A}\) is (a) \(43^{2}\) (b) \(2^{43}\) (c) \(43^{43}\) (d) \(2^{86}\)

When we talk about ordered pairs, we're referring to two elements grouped together in a specific sequence; the first component is called the 'first' and the second is called the 'second'. In mathematics, ordered pairs are written within parentheses, like this: (a, b). The concept of an ordered pair is fundamental in various areas, including set theory and coordinate geometry.

Ordered pairs have a crucial property: (a, b) is only equal to (c, d) if and only if a equals c and b equals d. This means that the order in which the elements are arranged matters—(a, b) is not the same as (b, a) unless a and b are equal.

An example can be seen in the Cartesian product of sets, where every element from one set is paired with every element from another set to create these pairs. For instance, if you have set A = {1,2} and set B = {3,4}, the Cartesian product A × B would be {(1, 3), (1, 4), (2, 3), (2, 4)}. Easy, right? This concept was applied in the problem to calculate the number of elements common to each of the sets A × B and B × A, based on the shared elements.

The commutative property is a basic rule in mathematics that applies to many operations, including addition and multiplication. This property states that the order in which two numbers are added or multiplied does not change the result. For addition, for instance, 3+5 is the same as 5+3; similarly, 2×4 equals 4×2.

In the context of set theory and, more specifically, the Cartesian product, the commutative property generally does not apply—the ordered pairs (a, b) and (b, a) are distinct unless a is the same as b. However, when discussing the number of ordered pairs, the situation is different. An example from our exercise: even though the ordered pairs from A × B are not the same as those from B × A, the number of unique pairs you can create from the 43 common elements would indeed be the same. That's because for counting purposes, the pairing of elements (a, b) from A with B is equal in quantity to the pairing of elements (b, a) from B with A. So, the commutative property for enumeration holds in this specific case, which helped us reach the solution.

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. Sets can be anything: numbers, letters, colors, or even other sets. Set theory forms the foundation for much of mathematics; it's used to define and understand mathematical concepts such as relations, functions, and numbers themselves.

Key concepts within set theory include subsets, unions, intersections, and Cartesian products. In our problem, the focus was on intersecting sets and the Cartesian product. The intersection of sets A and B was given—43 shared elements, which we denoted as set C. Understanding the properties of set C and how the Cartesian product operates allowed us to calculate the number of ordered pairs for A × B and B × A. This applied knowledge exemplifies set theory's practical usage and shows how it can simplify complex problems by breaking them down into the behavior of sets and their elements.