Set Theory Lecture 4 Flashcards
What is an equivalence relation?
A relation that is reflexive, symmetric, and transitive.
What are the three properties of an equivalence relation?
Reflexivity, symmetry, and transitivity.
What is the reflexivity property of an equivalence relation?
For all x in A, x R x.
What is the symmetry property of an equivalence relation?
If x R y, then y R x.
What is the transitivity property of an equivalence relation?
If x R y and y R z, then x R z.
What is an example of an equivalence relation?
Congruence modulo m in integers.
What is the definition of congruence modulo m?
x ≡ y (mod m) if y - x is a multiple of m.
Why is congruence modulo m an equivalence relation?
It satisfies reflexivity, symmetry, and transitivity.
What are examples of congruence modulo m?
16 ≡ 4 (mod 12), 11 ≡ 25 (mod 7).
What is logical equivalence in propositional logic?
Two formulas φ and ψ are equivalent if they have the same truth value in all cases.
What are examples of logical equivalences?
De Morgan’s laws, distributivity, and contradictions.
What is an equivalence class?
The set of all elements related to a given element under an equivalence relation.
What is the notation for an equivalence class?
[a] = {x ∈ A : x ≡ a}.
What is an example of equivalence classes in binary numbers?
Grouping 3-bit numbers with the same number of 0s.
How do equivalence relations relate to partitions?
An equivalence relation naturally partitions a set into disjoint equivalence classes.
What is an example of a partition induced by an equivalence relation?
The set of integers partitioned into even and odd numbers.
What is a complete system of representatives?
A subset of A that contains exactly one element from each equivalence class.
What is an example of a complete system of representatives?
For congruence modulo 3, a possible set is {0,1,2}.
What is a theorem about complete systems of representatives?
A subset S is a complete system of representatives if every element in A is equivalent to some element in S and no two elements in S are equivalent.
What is an example of a set partitioned by equivalence classes?
The set {0,1}³ partitioned by the number of 0s.
What is an example of equivalence classes in modular arithmetic?
Congruence modulo 3 partitions integers into three sets: multiples of 3, numbers one greater than a multiple of 3, and numbers two greater than a multiple of 3.
What is an example of an equivalence relation on strings?
Two strings are related if they start with the same letter.
What is another example of an equivalence relation on strings?
Two strings are related if they contain the same number of a’s.
What does it mean for an equivalence relation to induce a partition?
Each equivalence class forms a distinct subset, and together they cover the entire set.
What is the significance of equivalence relations in mathematics?
They provide a formal way to group objects with similar properties.
What is the relationship between partitions and equivalence relations?
Every partition of a set corresponds to an equivalence relation, and vice versa.
What is an application of equivalence relations in computer science?
Memory management and hash table collision resolution.
What is an example of equivalence relations in real life?
Grouping people by birth year in a school system.
How do equivalence classes help in simplifying mathematical structures?
They allow us to group elements into representative categories, reducing complexity.
What is the connection between equivalence relations and logic?
Logical equivalence is an example of an equivalence relation, grouping logically identical statements together.
