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

Prove that cardinality is an equivalence relation. Prove for all properties (refextivity, transitivity and symmetry). Please do this problem and explain every step. The less confusing notation the better, thanks!

Expert Solution

Colby Messinger answered 2 years ago

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 ?∈?...

Prove the transitivity of similarity: If △ ???~ △ ??? and △ ???~ △ ??? then...

Prove the transitivity of similarity: If △ ???~ △ ??? and △ ???~ △ ??? then △ ???~ △ ???. Write this proof in the numbered statements and reasons style.and also a picture of triangles.

Prove that "congruent modulo 3" is an equivalence relation on Z. 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 \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)

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?

3) Prove that the cardinality of the open unit interval, (0,1), is equal to the cardinality...

3) Prove that the cardinality of the open unit interval, (0,1), is equal to the cardinality of the open unit cube: {(x,y,z) E R^3|0<x<1, 0<y<1, 0<Z<1}. [Hint: Model your argument on Cantor's proof for the interval and the open square. Consider the decimal expansion of the fraction 12/999. It may prove handdy]

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that...

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that X / R is a partition of X.

Prove that isomorphism is an equivalent relation on the set of all groups.

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