Show A ∩ B by Venn diagram. When A ⊆ B. Show by Venn diagram A ∩ (B ∪ C). Define intersection of two sets. Define a function. Define one-one function. Define an onto function. Defi... Show A ∩ B by Venn diagram. When A ⊆ B. Show by Venn diagram A ∩ (B ∪ C). Define intersection of two sets. Define a function. Define one-one function. Define an onto function. Define a bijective function. Write De Morgan's laws.
Understand the Problem
The question consists of multiple parts asking for definitions and demonstration of various mathematical concepts related to set theory and functions, including intersection, Venn diagrams, one-one, onto, and bijective functions, as well as De Morgan's laws.
Answer
Venn diagrams and definitions are provided, including De Morgan's laws.
The final answer is given below:
Venn Diagrams
- A ∩ B when A ⊆ B: Since A is a subset of B, A ∩ B is simply A.
- A ∩ (B ∪ C): Draw three circles intersecting. Shade the area that belongs to both A and B or C.
Definitions
- Intersection of Two Sets: A ∩ B = {x | x ∈ A and x ∈ B}.
- Function: A relation f from a set X to a set Y such that every element x in X is related to exactly one element y in Y.
- One-One Function (Injective): A function f: X → Y is one-one if different elements in X have different images in Y.
- Onto Function (Surjective): A function f: X → Y is onto if every element y in Y is the image of at least one element x in X.
- Bijective Function: A function is bijective if it is both one-one and onto.
De Morgan's Laws
- (A ∪ B)' = A' ∩ B'
- (A ∩ B)' = A' ∪ B'
Answer for screen readers
The final answer is given below:
Venn Diagrams
- A ∩ B when A ⊆ B: Since A is a subset of B, A ∩ B is simply A.
- A ∩ (B ∪ C): Draw three circles intersecting. Shade the area that belongs to both A and B or C.
Definitions
- Intersection of Two Sets: A ∩ B = {x | x ∈ A and x ∈ B}.
- Function: A relation f from a set X to a set Y such that every element x in X is related to exactly one element y in Y.
- One-One Function (Injective): A function f: X → Y is one-one if different elements in X have different images in Y.
- Onto Function (Surjective): A function f: X → Y is onto if every element y in Y is the image of at least one element x in X.
- Bijective Function: A function is bijective if it is both one-one and onto.
De Morgan's Laws
- (A ∪ B)' = A' ∩ B'
- (A ∩ B)' = A' ∪ B'
More Information
The Venn diagrams visualize subsets and set operations. De Morgan's laws provide relationships between unions, intersections, and complements.
Tips
Be sure to accurately shade or label parts of the Venn diagram to avoid confusion.
Sources
- Sets and Functions - UC Davis Math - math.ucdavis.edu
- Venn Diagrams and set operations – Laws of set theory - mathssnsce.weebly.com
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