Problem 64 - Discrete Mathematics with Applications

Question

Simplify each set expression. $$ \left(A \cup B^{\prime}\right)^{\prime} \cap\left(A^{\prime} \cap B\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(A^{\prime} \cap B\).

Step 1: Apply De Morgan's Law on the first part of the expression (left side)

According to De Morgan's law, for any sets A and B: \[ (A \cup B)^{\prime} = A^{\prime} \cap B^{\prime} \] Applying this to \(\left(A \cup B^{\prime}\right)^{\prime}\): \[ \left(A \cup B^{\prime}\right)^{\prime} = A^{\prime} \cap (B^{\prime})^{\prime} \]

Step 2: Use the complement property on the result from Step 1

The complement property states that for any set A: \[ (A^{\prime})^{\prime} = A \] Applying this to \(A^{\prime} \cap (B^{\prime})^{\prime}\): \[ A^{\prime} \cap (B^{\prime})^{\prime} = A^{\prime} \cap B \]

Step 3: Substitute the results from Step 1 and 2 into the original expression

Replace \(\left(A \cup B^{\prime}\right)^{\prime}\) in the original expression with its simplified form, \(A^{\prime} \cap B\): \[ (A^{\prime} \cap B) \cap (A^{\prime} \cap B) \]

Step 4: Apply the idempotent law to simplify the expression

The idempotent law states that for any set A: \[ A \cap A = A \] Applying this to \((A^{\prime} \cap B) \cap (A^{\prime} \cap B)\): \[ (A^{\prime} \cap B) \cap (A^{\prime} \cap B) = A^{\prime} \cap B \] The simplified expression is: \[ A^{\prime} \cap B \]

Key Concepts

Set Theory

Set theory is a foundational system for mathematics, focusing on the study of sets, which are collections of objects. Understanding sets is crucial for numerous mathematical areas, including relations, functions, and probability.

In set theory, objects within a set are called elements or members. For example, if we have a set containing the numbers 1, 2, and 3, it can be represented as \( \{1, 2, 3\} \) where the curly braces denote a collection of distinct elements. Set theory provides the language and the tools to manipulate these sets in meaningful ways through various operations, such as union (\( \cup \) - combining elements from two sets), intersection (\( \cap \) - common elements between two sets), and complement (the elements not in a set).

These operations help to construct new sets from existing ones and are governed by rules and laws - like De Morgan's Laws, which describe the relationship between the union and intersection of sets and their complements.

Complement of Sets

The complement of a set is a fundamental concept in set theory and refers to the elements not included in the set, within a given universal set. If we consider all possible elements in a universal set \( U \), the complement of a set \( A \) (denoted as \( A^\prime \) or \( A^c \) ) is basically the 'opposite' of \( A \), meaning the set of all elements in \( U \) that are not in \( A \).

For example, if our universal set \( U \) is the set of all whole numbers, and \( A \) contains just the even numbers, then the complement of \( A \) would include all the odd numbers in \( U \).

An important property of complements is when you take the complement of a complement, you end up back where you started: \[ (A^\prime)^\prime = A \]This principle plays a pivotal role in understanding and navigating through more complex set operations and is a part of the solution process for the given exercise.

Idempotent Law

The idempotent law is a principle within set theory that has a straightforward yet powerful implication. It pertains to operations performed on a set that, no matter how many times they are applied, yield the same result as if they were done once.

Specifically, in set theory, the idempotent law applies to the intersection and union operations. When you intersect a set with itself \( A \cap A \), or unite a set with itself \( A \cup A \), the result is simply the set \( A \).

Intersection Example

\( A \cap A = A \)

Union Example

\( A \cup A = A \)
These operations don't change the set. This law is applied in the textbook exercise's final step after utilizing De Morgan's Laws and the complement property, ultimately simplifying the complex set expression to its most basic form.