Podcast
Questions and Answers
Which of the following statements about set operations is true?
Which of the following statements about set operations is true?
- A ∩ B is disjoint from A ∪ B
- A Δ B is a subset of A ∪ B (correct)
- A ∪ B is always disjoint from A ∩ B
- A and B must be equal for A ∪ B to equal A ∩ B
A Δ B is equivalent to A ⊕ B.
A Δ B is equivalent to A ⊕ B.
True (A)
What can be used to disprove statements about sets?
What can be used to disprove statements about sets?
Counterexample
If A = {1, 2, 3} and B = {2, 3, 4}, then A Δ B = __________.
If A = {1, 2, 3} and B = {2, 3, 4}, then A Δ B = __________.
Match the following terms with their descriptions:
Match the following terms with their descriptions:
Which scenario can be used to disprove the statement: ∀A, B, C ⊆ U, A ∩ C ⊆ B ∩ C ∧ A ∪ C ⊆ B ∪ C ⇒ A = B?
Which scenario can be used to disprove the statement: ∀A, B, C ⊆ U, A ∩ C ⊆ B ∩ C ∧ A ∪ C ⊆ B ∪ C ⇒ A = B?
The symmetric difference A Δ B can include elements that are present in both A and B.
The symmetric difference A Δ B can include elements that are present in both A and B.
What does it mean if A ∪ B ⊆ A ∩ B?
What does it mean if A ∪ B ⊆ A ∩ B?
Flashcards
Truth Table Proof for Sets
Truth Table Proof for Sets
A proof technique that uses truth tables to demonstrate the validity of a conditional statement involving sets.
Counterexample Proof for Sets
Counterexample Proof for Sets
A proof technique that uses a specific example to demonstrate the falsity of a universal statement involving sets.
Symmetric Difference (A Δ B)
Symmetric Difference (A Δ B)
The set of elements that are in A or in B, but not in both.
Intersection (A ∩ B)
Intersection (A ∩ B)
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Union (A ∪ B)
Union (A ∪ B)
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Difference (A - B)
Difference (A - B)
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Direct Proof for Sets
Direct Proof for Sets
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Proof by Contradiction for Sets
Proof by Contradiction for Sets
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Study Notes
Set Proofs and Disproofs
- Set proofs and disproofs can utilize various proof techniques.
- Example: Proving VA, B ⊂ U, A ∪ B = A ∩ B → A = B
- Proof method involves assuming x ∈ A
- x ∈ A ∪ B (generalization)
- x ∈ A ∩ B (assumption)
- x ∈ A or x ∈ B (definition of ∪)
- x ∈ A and x ∈ B (definition of ∩)
- x ∈ B (from the previous step)
- Thus, B ⊆ A (generalization and specification)
- Similarly, A ⊆ B, thus A = B
Conditional Statements via Truth Tables
- Truth tables can be used to prove conditional statements involving sets.
- Example A, B ⊂ U, A ∪ B ∩ A ∩ B = A ∪ B
- Truth table method involves evaluating all possible combinations of A, B values.
Disproofs via Counterexamples
- Counterexamples can be used to disprove statements involving sets.
- Example: Disproving A, B, C ⊂ U, (A ∩ C) ∩ (B ∩ C) = A ∩ B ∩ C
- Visualization using Venn diagrams helps find counterexamples.
Symmetric Difference
- Symmetric difference (A Δ B) is defined as (A - B) ∪ (B - A).
- Example: Given A = {1, 2, 3}, B = {2, 3, 4}, find A Δ B.
Proof using Set Theory
- Proving statements involving sets might involve demonstrating one set is a subset of another.
- The elements of the subset must be present in the set.
Power Sets
- Example: If A and B are subsets of the universal set (U), proving P(A) ∩ P(B) = P(A ∪ B).
- Proof: Demonstrating a set element is in the power set of A (x ∈ P(A)).
- x is present in the power set of B (x ∈ P(B)).
- Therefore x is in the power set of A ∪ B as well (x ∈ P(A ∪ B)).
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