Problem 40 Let \(A\) be a set with 12 eleme... [FREE SOLUTION] (2024)

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When dealing with sets in mathematics, a crucial concept to understand is the power set. The power set of any given set A is a larger set that contains all possible subsets of A, including the empty set and A itself. For example, if we imagine a simple set containing three elements, say {a, b, c}, its power set would include all combinations from no elements (the empty set) to all elements ({a, b, c}).

To find out how many subsets a set has, we can use the formula related to the power set, which states that if a set has 'n' elements, the number of subsets (the size of the power set) is given by the mathematical expression \(2^n\). This is because each element can either be included in or excluded from a subset, effectively giving us two options per element. So, when a set has 12 elements, like in our exercise, by raising 2 to the 12th power, we get \(2^{12}\) which equals 4096. This includes every possible subset, from the one with zero elements (empty set) to the one with all 12 elements.

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. It is the foundational basis for many areas of mathematics. Sets can be anything: numbers, people, letters, etc. For instance, a set A with 12 elements could represent 12 different numbers or 12 unique letters of the alphabet.

The beauty of set theory lies in its ability to apply uniform rules to these collections, such as finding the number of subsets as seen in the exercise. In set theory language, a subset is a set where all of its elements are contained within another set. Importantly, under set theory, the empty set is considered a subset of any set, contributing to the total count when we calculate the power set. It's also worth noting that the principles of set theory are widely applied across various fields such as statistics, computer science, and logic.

Combinatorics is a field of mathematics that deals with counting, arrangement, and combination of elements within sets, proving essential for solving our exercise. It comes into play, particularly when finding the number of subsets with certain properties – such as those with exactly one element or two or more elements.

In our exercise, combinatory logic was used to determine different types of subsets. For instance, single-element subsets are directly related to the number of elements in the set. Since set A has 12 elements, there are 12 possible subsets each containing exactly one element. On the other hand, for subsets with two or more elements, we consider the full range of subsets first and exclude those that do not meet our criteria, which in this case are the single-element subsets and the empty set. Therefore, combinatory methods streamline the process of counting and organizing such subsets.

Mathematical induction is a powerful proof technique used to establish the truth of an infinite sequence of propositions. While it's not explicitly employed in our exercise, understanding it can enhance the grasp of how certain mathematical truths are concluded, such as the formula for the number of subsets of a set (\(2^n\)).

Inductive reasoning starts with a base case, where the proposition is shown to be true. Then, it assumes that the proposition is true for some arbitrary case n, and on that basis, proves it is true for the next case n+1. This establishes a domino effect, confirming the proposition for all natural numbers. For the subsets of a set, one could use mathematical induction to show why the power set formula is always valid, thus cementing the basis for many combinatorial calculations.