Set Operations – Explanation, Properties, Subset and Powerset

Set Operations – Explanation, Properties, Subset and Powerset

What are Set Operations?

Set operations are mathematical operations that are used to combine two sets into a new set. The most common set operations are union, intersection, and difference.

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The Union of Sets (∪)

The union of sets is a mathematical operation that combines two sets into a single set. The union is denoted by the symbol ∪. The union of two sets A and B is the set of all elements that are in either A or B, or both A and B.

For example, the union of the sets {1, 2, 3} and {4, 5, 6} is the set {1, 2, 3, 4, 5, 6}. The union of the sets {1, 2} and {3, 4} is the set {1, 2, 3, 4}.

The Intersection of Sets (∩)

The intersection of sets A and B is the set of all elements that are in both A and B. Symbolically, this is written as A ∩ B.

For example, if A = {1, 2, 3} and B = {4, 5, 6}, then A ∩ B = {4, 5, 6}.

The Difference Between Sets (-) and Subsets (-)

A set is a collection of objects, while a subset is a specific type of set that has a specific number of elements. Sets can be composed of anything, while subsets must have a specific, predetermined number of elements.

Venn Diagram for Union of Sets

A Venn diagram is a graphical representation of the relationships between sets. In a Venn diagram, each set is represented by a circle, and the circles are overlapping or intersecting. The relationships between the sets are represented by the lines connecting the circles.

In the diagram, the blue circle represents the set of all numbers that are less than 10, the purple circle represents the set of all numbers that are greater than 10, and the orange circle represents the set of all numbers that are equal to 10. The lines between the circles indicate that the sets are intersecting.

A and B

A B

A ∩ B

A∪ B
=

A
∩ (B
∪ C)
=

A
∩ (B
∩ C)
=

A
∪ (B
∩ C)
=

A
∪ (B
∪ C)
=

A
∪ (B
∪ C)
=

A
∩ (B
∪ C)
=

Venn Diagram for Difference Between Sets

A

B

A is not in B

A is a subset of B

Properties of Set Operations

The following properties always hold for set operations:

1. Commutativity: A ∪ B = B ∪ A and A ∩ B = B ∩ A.

2. Associativity: A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C.

3. Distributivity: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

4. De Morgan’s laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

What is a Subset?

A subset is a group of items that are part of a larger group. The subset is a smaller group within a larger group.

What is Powerset?

Powerset is a search engine for the web that uses natural language processing to help you find information.

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