Complements of Sets

Blue Shirt Khaki Pants
8 Jul 201402:25

Summary

TLDRThis 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).

Outlines

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الوسوم ذات الصلة
Set TheoryComplement SetsUniversal SetMath LessonPropertiesSubsetNull SetUnionElementsEducation
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