Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the pairs of sets are disjoint. a. \(\varnothing,\\{1,3,5\\}\) b. \(\\{0,1,3,4\\},\\{0,2,5,7\\}\)

Step-by-Step Solution

Answer

The first pair of sets, \(\varnothing\) and \(\{1,3,5\}\), is disjoint as the empty set has no elements. The second pair of sets, \(\{0,1,3,4\}\) and \(\{0,2,5,7\}\), is not disjoint since they have the common element 0.

Step 1: Identify the Pairs of Sets

There are two pairs of sets given: a. \(\varnothing,\\{1,3,5\\}\) b. \(\\{0,1,3,4\\},\\{0,2,5,7\\}\)

Step 2: Check for Common Elements in the First Pair of Sets

In the first pair of sets, one set is the empty set (\(\varnothing\)) which contains no elements, and the other set is \(\{1,3,5\}\). Since the empty set has no elements, it cannot have any common elements with any other set. Therefore, the first pair of sets is disjoint.

Step 3: Check for Common Elements in the Second Pair of Sets

In the second pair of sets, we have: Set A = \(\{0,1,3,4\}\) and Set B = \(\{0,2,5,7\}\) To determine if these sets are disjoint, we need to check if they have any common elements. We can see that both Set A and Set B have the element 0. Since they have a common element, the second pair of sets is not disjoint. In conclusion, the first pair of sets is disjoint, while the second pair of sets is not disjoint.

Key Concepts

Empty Set

The concept of the empty set, often denoted by the symbol \( \varnothing \), is a foundational idea in set theory. Simply put, the empty set is a set that contains no elements at all. It's like an empty basket, where there's nothing inside to count or to look at. This concept might seem simple, but it plays a crucial role in mathematics.

• In comparison with other sets, the empty set is unique because no matter what element you consider, it is never part of the empty set.
• The empty set is a subset of every set, including itself. This may sound strange, but it adheres to the rule that says a set A is a subset of a set B if every element of A is also an element of B.

When considering whether pairs of sets are disjoint, as in the exercise, if one of the sets is an empty set, then this pair is automatically disjoint. That's because there are no elements in an empty set to share with the other set.

Common Elements

In order to decide if two sets are disjoint, we need to determine whether they share any common elements. Common elements are simply those elements that appear in both sets.

• If any single element can be found in both sets, then the sets are not disjoint since they have something in common.
• If there are no common elements, then the two sets are disjoint.

For example, in the second pair of sets from the exercise, Set A \(\{0,1,3,4\}\) shares the element "0" with Set B \(\{0,2,5,7\}\). Because of this common element, these two sets are not disjoint. Checking for shared elements requires systematically comparing each element from one set with all elements from the other set. It can be a straightforward process if the sets are small, but it becomes more involved with larger sets.

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. These objects can be anything: numbers, letters, or even other sets. The simplicity and flexibility of set theory make it an important part of mathematics.

• In set theory, sets are the building blocks for more complex mathematical concepts, such as relations, functions, and more.
• Understanding the nature of sets, like the empty set and the notion of common elements, is important for solving problems concerning disjoint sets.

The concept of disjoint sets, for instance, stems from the fundamental idea of comparing elements in different sets to see if there's any overlap. This principle is used to define relationships between sets and measure the extent to which they intersect or remain separate. Studying sets can help students grasp abstract concepts and see the logical structures underlying mathematics.