Question

Suppose \(\left\\{\begin{aligned} A_{1} &=\\{0,2,4,8,10,12,14,16,18,20,22,24\\}, \\ A_{2} &=\\{0,3,6,9,12,15,18,21,24\\}, \\ A_{3} &=\\{0,4,8,12,16,20,24\\} . \end{aligned}\right.\) (a) \(\bigcup_{i=1}^{3} A_{i}=\) (b) \(\bigcap_{i=1}^{3} A_{i}=\)

Short answer

\(\bigcup_{i=1}^{3} A_{i}=\{0,2,3,4,6,8,9,10,12,14,15,16,18,20,21,22,24\}\) and \(\bigcap_{i=1}^{3} A_{i}=\{0,12,24\}\)

Step-by-step solution

Union of Sets

First, identify all the unique elements present in the sets \(A_1, A_2, A_3\) and merge them together. The result will be the union of these sets.\nThe union of \(\bigcup_{i=1}^{3} A_{i}=\{0,2,3,4,6,8,9,10,12,14,15,16,18,20,21,22,24\}\)

Intersection of Sets

Notice which elements are commonly present in all three sets \(A_1, A_2, A_3\). These common elements will form the intersection of these sets. The intersection is \(\bigcap_{i=1}^{3} A_{i}=\{0,12,24\}\)

Key concepts

Union of Sets

When we talk about the union of sets, we refer to the process of combining all the elements from the given sets into a new set which contains every distinct element from each of the original sets. This operation is symbolized using the union symbol '∪'. When you perform the union of two or more sets, you include all elements from each set, but you eliminate any duplicates to ensure that each element is unique.

For example, let's consider the sets given in the exercise. The set \( A_1 \) includes even numbers between 0 and 24, while \( A_2 \) contains multiples of three in the same range. \( A_3 \) is made up of even numbers skipping every second one. When creating the union of these sets, \( \bigcup_{i=1}^{3} A_{i} \), you'll notice that some numbers are repeated across the sets. By merging them and removing these repetitions, you get the complete set of unique numbers that appear in any of the three sets. This is useful in various fields, such as database queries where you may want to combine different data points without any redundancy.

Intersection of Sets

The intersection of sets captures the idea of commonality. It is the set containing all the elements that are shared by all the given sets. The symbol '∩' is used to represent this operation. The intersection is interesting because it tells us what the sets have in common—an element needs to be in each set to be part of the intersection. This process effectively 'filters out' any elements that are not found in every single set you are considering.

In the context of our example, \( A_1, A_2, \) and \( A_3 \) have some numbers that are present in all three. By analyzing each set, we see that these numbers are 0, 12, and 24, which form the set \( \bigcap_{i=1}^{3} A_{i} \). Intersection operations are particularly crucial in areas such as survey analysis, where you might want to find common preferences among different demographic groups, or in logic, where intersections help determine if arguments are mutually consistent.

Elementary Set Operations

Elementary set operations include the basic processes you can perform on sets, such as union, intersection, difference, and complement. These operations provide a foundation for set theory, a mathematical discipline that examines how groups of objects interact with one another.

Difference and Complement

Excluding the union and intersection which we've already covered, the difference of sets is the set of elements that belong to one set and not another—symbolized as '−'. The complement, symbolized by a bar or a prime mark over the set name, consists of all elements not in the set but within a larger universe of discourse.

These operations are not just academic; they have practical implications in fields such as computer science for data structuring, in probability theory for determining outcomes of certain events, and in everyday problem-solving. Understanding these foundational concepts is crucial for advancing in more complex areas of mathematics and its applications in various scientific disciplines.