Problem 22 - Discrete Mathematics with Applications

Question

Using the sets \(A=\\{a, b, e, h\\}, B=\\{b, c, e, f, h\\}, C=\\{c, d, f, g\\},\) and \(U=\\{a, \ldots, h\\},\) find the binary representation of each set. \(A \cup(B \cap C)\)

Step-by-Step Solution

Answer

The binary representation for \(A \cup (B \cap C)\) based on the universal set U is 11011001.

Step 1: Find B ∩ C

To find the intersection of sets B and C, list out the elements that are common to both sets. In this case, B ∩ C = {c, f}.

Step 2: Find A ∪ (B ∩ C)

Now let's find the union of set A and the result from Step 1 (B ∩ C). To find the union, list out all the elements in either set A or in (B ∩ C) without duplication. A ∪ (B ∩ C) = {a, b, e, h, c, f}.

Step 3: Identify the Universal Set

The universal set U is given as {a, ..., h}. The elements in the universal set are in alphabetical order.

Step 4: Convert A ∪ (B ∩ C) to Binary Representation

To convert A ∪ (B ∩ C) to binary representation, write a 1 for each element present in A ∪ (B ∩ C) and a 0 for each element not present, using the order from the universal set U. For A ∪ (B ∩ C) = {a, b, e, h, c, f}, the binary representation is 11011001. So, the binary representation for A ∪ (B ∩ C) based on the given universal set U is 11011001.

Key Concepts

Union of Sets

In set theory, the union of sets refers to the combination of all unique elements from two or more sets. When you want to find the union, you pick each element that appears in any of the involved sets.

For example, consider the sets \(A = \{a, b, e, h\}\) and \(B = \{b, c, e, f, h\}\). To calculate \(A \cup B\), you take all the distinct elements present in set \(A\), then add any elements in set \(B\) that are not yet in the union. So, \(A \cup B = \{a, b, e, h, c, f\}\).

Remember, union means inclusivity: you include everything from all sets, but don't repeat any item more than once.

Intersection of Sets

The concept of intersection in set theory refers to finding common elements between two or more sets. This is useful for identifying shared characteristics.

For example, for sets \(B = \{b, c, e, f, h\}\) and \(C = \{c, d, f, g\}\), the intersection \(B \cap C\) contains only elements found in both sets. In this case, \(B \cap C = \{c, f\}\).

Think of intersection as the overlap where the sets meet. It's like finding all the items that two friends share in their personal collections.

Universal Set

The universal set is a fundamental concept in set theory, representing the set that contains all possible elements within a particular context or discussion. It acts as a master set.

In the given exercise, the universal set \(U\) is \(\{a, b, c, d, e, f, g, h\}\). This means any set you discuss using this universal set will always have elements from this specific group.

The universal set helps in defining subsets, complements, and binary representations, providing a complete framework for the analysis.

Binary Representation of Sets

Binary representation of sets provides a way to express subsets of a universal set using binary numbers. In this representation, each element in the universal set is assigned a digit.

Here's how you use it with \(A \cup (B \cap C)\), which was found to be \(\{a, b, e, h, c, f\}\). Represent elements as 1 if they are in this union and 0 if they aren't, following the order of \(U = \{a, b, c, d, e, f, g, h\}\). Thus, the binary code becomes 11011001.

- '1' signifies presence in the set.
- '0' signifies absence.

Binary representation is especially helpful in computer science for data manipulation and easy comparison of sets.