Complements of Sets

Summary

This lesson in Unit 5.3 covers the concept of the complement of sets. Starting with a universal set (U) and a subset (A), the complement of A, denoted as A prime, includes elements in U but not in A. The video explains key properties of complements: the intersection of A and A prime is empty, their union equals U, and the number of elements in A plus A prime equals the total in U. An example uses letters of the alphabet to illustrate these principles. The lesson is designed to clarify these set operations with practical examples.

Takeaways

• 🔢 The lesson focuses on the concept of complements of sets.
• 📋 A complement is defined as the set of elements in the universal set but not in subset A.
• 🌐 The universal set is represented by U and subset A is part of U.
• 📝 A prime (A') refers to the complement of A, symbolized with a tick mark.
• 🧮 Example: If U = {1, 2, 3, 4, 5} and A = {1, 3}, A' = {2, 4, 5}.
• ⚖️ The intersection of A and A' is the null set, meaning no common elements.
• 🔄 The union of A and A' returns the entire universal set U.
• 🔢 The sum of elements in A and A' equals the total number of elements in U.
• 🅰️ An example using the alphabet: A = letters in 'ambidextrous', A' = remaining letters in the alphabet.
• ✅ A and A' have no common elements and their union covers the entire alphabet.

Q & A

What is a complement of a set?

The complement of a set A, written as A prime (A'), is the set of elements that are in the universal set U but not in set A.

What symbol is used to denote the complement of a set?

The complement of a set is denoted by a prime symbol (A'), which is a small tick mark next to the set's name.

How is the universal set represented?

The universal set is represented by the symbol 'U' and contains all the possible elements under consideration.

Can you provide an example of a set and its complement?

If U is the set {1, 2, 3, 4, 5} and A is the subset {1, 3}, then A' would be the set {2, 4, 5}, as these are the elements in U that are not in A.

What is the intersection of a set and its complement?

The intersection of a set and its complement is the null set (∅), meaning they have no elements in common.

What is the union of a set and its complement?

The union of a set and its complement is the universal set U, as combining both gives back all the elements in U.

How do the number of elements in a set and its complement relate to the universal set?

The number of elements in a set A plus the number of elements in its complement A' equals the number of elements in the universal set U.

Can you give an example with letters of the alphabet?

If U is the set of all letters in the alphabet and A is the set of letters in the word 'ambidextrous', then A' would be the letters not in the word 'ambidextrous'. Together, A and A' contain all the letters of the alphabet with no overlap.

What are the properties of the complement of a set?

The properties of a complement are: (1) The intersection of a set and its complement is the null set, (2) The union of a set and its complement is the universal set, and (3) The number of elements in a set and its complement together equals the total number of elements in the universal set.

What is the relationship between the set 'ambidextrous' and its complement?

The set A of letters in 'ambidextrous' has 12 elements, and its complement A' has 14 elements, making a total of 26 elements, which corresponds to the total number of letters in the alphabet (U).