Define a subset and give one example. Write all the subsets of the set {a, b}. Show A ∩ B by Venn diagram. When A ⊆ B. Show by Venn diagram A ∩ (B ∪ C). Define intersection of two... Define a subset and give one example. Write all the subsets of the set {a, b}. 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.
Understand the Problem
The question is asking for definitions and examples related to sets and functions, specifically subsets, intersections, and types of functions. Each point requires a specific answer or demonstration, such as using Venn diagrams for visual aids.
Answer
A subset is a set with elements from another set, e.g., \( A \subseteq B \). Subsets of \(\{a, b\}\) are \(\{\}, \{a\}, \{b\}, \{a, b\}\). \( A \cap B \) shows \( A \) inside \( B \). An intersection is \( A \cap B \). A function assigns one output to each input. A one-one function has unique outputs, while an onto function covers all elements in the codomain.
Answer for screen readers
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A subset is a set where all elements are contained in another set. Example: ( A = {1, 2} \subseteq B = {1, 2, 3} ).
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The subsets of ({a, b}) are ({}, {a}, {b}, {a, b}).
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Venn diagram for ( A \cap B ) shows ( A ) inside ( B ) with ( A \cap B = A ).
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Venn diagram for ( A \cap (B \cup C) ) illustrates ( A ) overlapping both ( B ) and ( C ).
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Intersection ( A \cap B ) is the set of common elements.
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A function ( f: X \rightarrow Y ) assigns one output for each input.
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A one-one function ensures ( f(x_1) = f(x_2) \implies x_1 = x_2 ).
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An onto function covers all elements in the codomain.
Steps to Solve
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Define a Subset A subset is a set where every element in the subset is also contained in another set. For example, if ( A = {1, 2} ) and ( B = {1, 2, 3} ), then ( A \subseteq B ).
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List All Subsets of ({a, b}) The subsets of ({a, b}) can be listed as follows:
- The empty set: ({})
- The single element subsets: ({a}), ({b})
- The full set: ({a, b})
So, all subsets are ({}, {a}, {b}, {a, b}).
- Draw Venn Diagram for ( A \cap B ) When ( A \subseteq B ) In this case, we visualize two circles where one is entirely within the other. Let ( A = {1, 2} ) and ( B = {1, 2, 3} ).
- In the Venn diagram, circle A will be inside circle B.
- The intersection ( A \cap B = A ), which is ({1, 2}).
- Draw Venn Diagram for ( A \cap (B \cup C) ) To represent ( A \cap (B \cup C) ):
- Draw circles for ( B ) and ( C ) with some overlap.
- Circle ( A ) overlaps with the area corresponding to ( B ) and ( C ).
- The intersection will be the part where circle ( A ) overlaps with either circle ( B ) or circle ( C ).
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Define Intersection of Two Sets The intersection of two sets ( A ) and ( B ) is the set of elements that are common to both sets. It's denoted as ( A \cap B ).
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Define a Function A function is a relation that assigns exactly one output for each input from a given set. If ( f ) is a function that maps elements ( x ) to elements ( y ), we write it as ( f: X \rightarrow Y ).
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Define a One-One Function A one-one function, or injective function, is defined such that each element of the domain maps to a unique element in the codomain. If ( f(x_1) = f(x_2) ), then ( x_1 = x_2 ).
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Define an Onto Function An onto function, or surjective function, is defined as a function where every element in the codomain has at least one element from the domain mapped to it. In other words, the range of the function is equal to the codomain.
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A subset is a set where all elements are contained in another set. Example: ( A = {1, 2} \subseteq B = {1, 2, 3} ).
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The subsets of ({a, b}) are ({}, {a}, {b}, {a, b}).
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Venn diagram for ( A \cap B ) shows ( A ) inside ( B ) with ( A \cap B = A ).
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Venn diagram for ( A \cap (B \cup C) ) illustrates ( A ) overlapping both ( B ) and ( C ).
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Intersection ( A \cap B ) is the set of common elements.
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A function ( f: X \rightarrow Y ) assigns one output for each input.
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A one-one function ensures ( f(x_1) = f(x_2) \implies x_1 = x_2 ).
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An onto function covers all elements in the codomain.
More Information
Understanding these definitions helps in exploring higher concepts in set theory and functions, which are foundational in mathematics. Venn diagrams provide a visual understanding that aids in grasping these concepts clearly.
Tips
- Misunderstanding subsets: Remember that every set is a subset of itself and the empty set is a subset of every set.
- Confusing the different types of functions: Ensure to clarify the differences between injective, surjective, and bijective functions.
- Overlooking the meaning of intersection: Intersection only includes elements common to both sets.
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