Prove that "congruent modulo 3" is an equivalence relation on Z. What are the equivalence classes?

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Colby Messinger answered 6 days ago

7. Prove that congruence modulo 10 is an equivalence relation on the set of integers. What...

7. Prove that congruence modulo 10 is an equivalence relation on the set of integers. What do the equivalence classes look like?

Prove that the equivalence classes of an equivalence relation form a partition of the domain of...

Prove that the equivalence classes of an equivalence relation form a partition of the domain of the relation. Namely, suppose ? be an equivalence relation on a set ? and define the equivalence class of an element ?∈? to be [?]?:={?∈?|???}. That is [?]?=?(?). Divide your proof into the following three peices: Prove that every partition block is nonempty and every element ? is in some block. Prove that if [?]?∩[?]?≠∅, then ???. Conclude that the sets [?]? for ?∈?...

a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse modulo...

a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse modulo n is well defined as a residue class in Zn, and depends only on the residue class a in Zn. b) Let Z × n ⊆ Zn be the set of invertible residue classes modulo n. Prove that Z × n forms a group under multiplication. Is this group a subgroup of Zn? c) List the elements of Z × 9 . How many...

Prove: There are inﬁnitely many primes congruent to 3 modulo 8. Hint: Consider N = (p1p2···pr)2...

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The equivalence relation on Z given by (?, ?) ∈ ? iff ? ≡ ? mod...

The equivalence relation on Z given by (?, ?) ∈ ? iff ? ≡ ? mod ? is an equivalence relation for an integer ? ≥ 2. a) What are the equivalence classes for R given a fixed integer ? ≥ 2? b) We denote the set of equivalence classes you found in (a) by Z_5. Even though elements of Z_5 are sets, it turns out that we can define addition and multiplication in the expected ways: [?] + [?]...

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What are the equivalence classes for a ∗ b ∗ ? • How many equivalence classes are there? • Make each one into a state and show how one can construct a minimal deterministic finite automaton from them. • Explain how to choose the start state and accepting states and how to draw the arrows. • The resulting automaton is minimal for this language. How about for {a n b n : n ≥ 0}? What are the equivalence classes?

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R...

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R is an equivalence relation, that is, prove that it is reflexive, symmetric, and transitive. Determine the equivalence classes of this relation. What members are in the class [2]? How many members do the equivalence classes have? Do they all have the same number of members? How many equivalence classes are there? Question 2. Equivalence Relation 2 Consider the relation from last week defined as:...

Prove that cardinality is an equivalence relation. Prove for all properties (refextivity, transitivity and symmetry). Please...

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Prove that \strongly connected" is an equivalence relation on the vertex set of a directed graph

Let G be a simple graph. Prove that the connection relation in G is an equivalence...

Let G be a simple graph. Prove that the connection relation in G is an equivalence relation on V (G)

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