Set Theory Basics

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Questions and Answers

What is the total number of subsets for the set A = {w, x, y, z}?

16 (correct)

Every element of A = {2, 4, 6} is also an element of B = {1, 2, 4, 6}.

True

Name the universal set if A = {2, 4, 6} and B = {1, 2, 4, 6}.

{1, 2, 4, 6}

The proper subsets of set A = {w, x, y, z} exclude the ______.

set itself Signup and view all the answers

Match the following terms with their definitions:

Universal Set = The set containing all possible elements in consideration Subset = A set where every element is contained in another set Proper Subset = A subset that does not include every element of another set Set Intersection = The set of elements that are common to both sets Signup and view all the answers

What is the result of the union of sets A = {2, 3, 5, 7} and B = {1, 2, 3, 4, 5}?

{1, 2, 3, 4, 5, 6, 7} Signup and view all the answers

The intersection of disjoint sets results in a non-empty set.

False Signup and view all the answers

What is the complement of a set A in a universal set U?

The set of all elements in U that are not in A. Signup and view all the answers

The result of the set difference A - B is composed of elements in A that are not in _____

B Signup and view all the answers

Match the set operation with its correct definition:

Union = Combines elements from both sets Intersection = Elements common to both sets Set Difference = Elements in one set but not the other Complement = Elements not in the given set but in the universal set Signup and view all the answers

If A = {a, b} and B = {b, c}, what is A ∩ B?

{b} Signup and view all the answers

The union of sets A and B can contain elements that are only in A.

True Signup and view all the answers

What is the result of the intersection of sets A = {1, 2, 3} and B = {4, 5, 6}?

The empty set {}. Signup and view all the answers

If set A contains a prime number less than 10, its elements are 2, 3, 5, and _____

7 Signup and view all the answers

What does the expression A - B represent?

Elements in A that are not in B Signup and view all the answers

What is the result of the union of sets A and B if A = {1, 2, 3, ..., 45} and B = {3, 6, 9, 12, ..., 27}?

{1, 2, 3, 4, ..., 45, 6, 9, 12, ..., 27} Signup and view all the answers

The complement of a set includes all elements that are not in the universal set.

False Signup and view all the answers

Define the term 'universal set.'

The universal set is the set that contains all possible elements for a particular discussion or problem. Signup and view all the answers

The intersection of sets A and B is the set of elements that are members of both A and ___ .

B Signup and view all the answers

Match the following terms with their correct definitions:

Union = The set of elements in either set Intersection = The set of elements in both sets Complement = Elements not in the specified set Set Difference = Elements in the first set but not in the second Signup and view all the answers

What does the term 'cardinality of a set' refer to?

The total number of elements in a set Signup and view all the answers

An empty set is always a subset of any set.

True Signup and view all the answers

What is the set difference A - B if A = {1, 2, 3, 4, 5} and B = {2, 4}?

{1, 3, 5} Signup and view all the answers

The set of all even numbers from 1 to 20 is denoted as ___.

{2, 4, 6, 8, 10, 12, 14, 16, 18, 20} Signup and view all the answers

Which of the following represents the complement of set C = {2, 4, 6, ..., 18} within a universal set U = {1, 2, ..., 20}?

{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20} Signup and view all the answers

Study Notes

Sets Defined

Set A = {2, 4, 6} and Set B = {1, 2, 4, 6}.
Every element of A is found in B, making A a subset of B.
Set B contains at least one element (1) not present in A.

Subsets and Proper Subsets

Subsets of set A = {w, x, y, z} consist of:
- No element: { }
- One element: {w}, {x}, {y}, {z}
- Two elements: {w, x}, {w, y}, {w, z}, {x, y}, {x, z}, {y, z}
- Three elements: {w, x, y}, {w, x, z}, {w, y, z}, {x, y, z}
- Four elements: {w, x, y, z}
Total number of subsets = 16 (calculated as 2^n, where n = number of elements).

Types of Sets

Universal Set (U): Contains all possible elements relevant to a particular discussion.
Empty Set (Null Set): Denoted by { }, it contains no elements and is a subset of every set.

Set Operations

Intersection: Set of elements common to both sets (A ∩ B).
Union: Set of elements in either set (A ∪ B), combining all unique members.
Set Difference: Elements in one set but not the other (A - B).
Complement of a Set: Contains all elements in the universal set not included in the specified set.

Practice Exercises

Example sets for practice include:
- A = {1,2,3,...,45}
- B = {x | x is a multiple of 3, 1 < x < 30}
- C = {even numbers from 1 to 18}
The exercises may include calculating cardinals and performing operations like intersection and union on various sets.

Key Set Operations Examples

Given sets A = {a, b, c, d, e} and B = {e, g, k, a}:
- A ∪ B = {a, b, c, d, e, g, k}
- B ∪ C will involve inclusion from both sets.

Advanced Set Operations

Involving more than two sets:
- Operations like (A ∪ C) ∪ B demonstrate the combinatorial nature of set union.

Exercise Review

Reviewing intersections and unions with examples:
- A ∩ B and C ∩ B lead to finding common elements across sets.
- Full compound operations like (A ∩ B) ∪ C consolidate knowledge of set combinations.

Summary

Understanding sets, their properties, subsets, and various operations forms a foundation for higher-level mathematical concepts.
Practice with real set examples builds proficiency in recognizing and manipulating sets in problem-solving contexts.

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Description

Explore the relationships between sets A and B in this quiz on set theory. Determine whether all elements of A are in B and identify any elements in B that are not in A. Test your understanding of basic set operations.