Anonymous
Mar 25
Notes and illustrations on set theory ?
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Mar 25
Set theory is a branch of mathematical logic that studies sets, which are collections of objects. Sets are typically denoted by curly braces { } and contain elements separated by commas. For example, the set of all even numbers can be written as {2, 4, 6, 8, ...}.
Some key concepts in set theory include:
1. Union: The union of two sets A and B, denoted by A ∪ B, is the set of all elements that are in either A, B, or both. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.
2. Intersection: The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that are in both A and B. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}.
3. Complement: The complement of a set A, denoted by A', is the set of all elements that are not in A. For example, if A = {1, 2, 3} and the universal set is {1, 2, 3, 4, 5}, then A' = {4, 5}.
4. Subset: A set A is a subset of another set B, denoted by A ⊆ B, if every element of A is also an element of B. For example, if A = {1, 2} and B = {1, 2, 3}, then A ⊆ B.
5. Power set: The power set of a set A, denoted by P(A), is the set of all subsets of A, including the empty set and A itself. For example, if A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}}.
Illustrations can help visualize these concepts, such as Venn diagrams showing the relationships between sets and operations like union and intersection. Additionally, diagrams can be used to represent set elements and their relationships in a more abstract or symbolic way.
Some key concepts in set theory include:
1. Union: The union of two sets A and B, denoted by A ∪ B, is the set of all elements that are in either A, B, or both. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.
2. Intersection: The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that are in both A and B. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}.
3. Complement: The complement of a set A, denoted by A', is the set of all elements that are not in A. For example, if A = {1, 2, 3} and the universal set is {1, 2, 3, 4, 5}, then A' = {4, 5}.
4. Subset: A set A is a subset of another set B, denoted by A ⊆ B, if every element of A is also an element of B. For example, if A = {1, 2} and B = {1, 2, 3}, then A ⊆ B.
5. Power set: The power set of a set A, denoted by P(A), is the set of all subsets of A, including the empty set and A itself. For example, if A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}}.
Illustrations can help visualize these concepts, such as Venn diagrams showing the relationships between sets and operations like union and intersection. Additionally, diagrams can be used to represent set elements and their relationships in a more abstract or symbolic way.
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