Problem 5 - Discrete Mathematics with Applications

Question

Let \(A\) and \(B\) be two sets such that \(|A|=2 a-b,|B|=2 a,|A \cap B|=a-b\) and \(|U|=3 a+2 b .\) Find the cardinality of each set. \(A \cup B\)

Step-by-Step Solution

Answer

The cardinality of the union set A ∪ B is \(3a\).

Step 1: Understand the Formula

We know the cardinality of the union of two sets is given by: \(|A ∪ B| = |A| + |B| - |A ∩ B|\). Apply the provided information to find the cardinality of A ∪ B.

Step 2: Substitute the Values

Substituting the given values into the formula: \(|A ∪ B| = (2a - b) + (2a) - (a - b)\)

Step 3: Simplify the Expression

Simplify the expression on the right-hand side by combining like terms: \(|A ∪ B| = (2a - b) + (2a) - (a - b) = 2a - b + 2a - a + b\) \(|A ∪ B| = 2a + 2a - a + b - b = 3a\)

Step 4: Find the Cardinality of A ∪ B

Now we have found that the cardinality of A ∪ B is 3a.

Key Concepts

Union of Sets

In set theory, the union of two sets is a fundamental operation that combines all the elements from both sets, without repeating any elements. The symbol used for union is \( \cup \), and it signifies that we take every unique element that exists either in set A or in set B, or in both.

When visualizing the union of sets, picture two overlapping circles where each circle represents a set. The area covered by both circles together—including the overlapping region—is the union.

Example of Union

If we have Set A = {1, 2, 3} and Set B = {3, 4, 5}, their union, represented as \( A \cup B \), would be {1, 2, 3, 4, 5}. Notice that the element '3' is not repeated, as the union only counts unique elements.

Intersection of Sets

Conversely, the intersection of sets captures the idea of commonality between sets. Using the symbol \( \cap \), the intersection refers to all elements that are present in both sets.

If two sets have no elements in common, their intersection is the empty set, often denoted by \( \emptyset \) or \( \{ \} \).

Example of Intersection

Returning to our earlier sets, Set A = {1, 2, 3} and Set B = {3, 4, 5}, the intersection \( A \cap B \) would be the set {3}, as '3' is the only element both sets share.

Cardinality Formula

The term cardinality refers to the number of elements in a set. When finding the cardinality of the union of two sets, a special formula helps to prevent double-counting elements found in the intersection of the two sets.

The formula is:\[|A \cup B| = |A| + |B| - |A \cap B|\]
Here, |A \cup B| represents the cardinality of the union of sets A and B, |A| and |B| are the cardinalities of sets A and B, respectively, and |A \cap B| is the cardinality of their intersection.

Applying the Cardinality Formula

Using this formula prevents you from counting any shared elements twice. It hands you the unique count of all elements present when the two sets are combined. This equation is essential in the study of set theory and is widely used in statistics, probability, and computer science.

Set Theory

Set theory is a branch of mathematical logic that deals with collections of objects, known as sets, and is a fundamental part of modern mathematics. It provides the foundational language for virtually all mathematical theories and concepts.

Some key concepts in set theory include elements, which are the objects contained within a set, and operations like unions and intersections, which allow for the combination and comparison of different sets.

In the context of our problem, set theory provides the tools to efficiently quantify and characterize the elements within sets A and B, as well as their union and intersection, using cardinality and the associated formulas.