Problem 2 - Discrete Mathematics with Applications

Question

Find the cardinality of each set. The set of letters of the word TWEEDLEDEE.

Step-by-Step Solution

Answer

The cardinality of the set of unique letters in the word TWEEDLEDEE is 5, as the set is {T, W, E, D, L}.

Step 1: Write down the given word

The given word is TWEEDLEDEE.

Step 2: Identify and list the unique letters

Identify the unique letters from the given word, remembering that each letter is counted only once, even if it appears multiple times. For the word "TWEEDLEDEE", the unique letters are {T, W, E, D, L}.

Step 3: Count the unique letters

Count the number of unique letters in the set. There are 5 unique letters: {T, W, E, D, L}.

Step 4: Write down the cardinality of the set

The cardinality of the set of unique letters in the word TWEEDLEDEE is 5.

Key Concepts

Unique Letters

When examining a word or phrase, you might be tasked with identifying unique letters. These are the letters that appear at least once in the word without taking into account repeated occurrences. For example, the word "TWEEDLEDEE" contains ten letters, but only some of them are unique. To find the unique letters:

• Break down the word letter by letter.
• List each distinct letter you encounter only once.

In "TWEEDLEDEE,” you’ll notice letters like 'E' appear multiple times. We count it once. Following this approach, the unique letters in this word are T, W, E, D, and L.

Set Theory

Set Theory is a branch of mathematical logic that focuses on the concepts of sets, which are collections of objects. In our context:

• A set is a collection of unique letters from the word.
• Sets are denoted with curly brackets like so: \( \{T, W, E, D, L\} \).

The primary concept in set theory that we deal with here is the "cardinality" of a set, which signifies the number of unique elements within it. We often use sets in problems like this to handle distinct members, which in our case, are the unique letters. This makes it easier to perform operations, such as counting them, by ignoring duplicates.

Counting

Counting is a fundamental operation in mathematics and is used extensively in set theory. In problems involving sets of letters:

• We count the number of unique items in a set to determine its cardinality.
• To do this, simply tally up how many distinct letters are in our set.

For the word "TWEEDLEDEE", after identifying all the unique letters (\( \{T, W, E, D, L\} \)), we see there are five distinct letters. Hence, counting gives us the cardinality of the set, which is 5. This process helps ensure accuracy by providing a step-by-step approach to determine the exact number of unique elements in any set.