Equivalence classes

Let p, p' be equivalence relations on a set A. Let n, n' be the number of equivalence classes pf p, p' respectively.... Please look at the attached doc for rest of question.

Prove the Transitive theory

Prove the following theory: 1) R1 is a subset of R2 => All of R3, R1R3 is a subset of R2R3 and 2) R1 is a subset of R2 => All of n, (R1)^n is a subset (R2)^n 3) Suppose R is transitive, then for all of n, R^n is a subset of R.

Partitions on a set

We denote the number of partitions of a set of n elements by P(n). Suppose the number of partitions of a set on n elements into k parts is denoted by P(n,k). Then obviously P(n) = P(n,1) + P(n,2) + ….. + P(n,n) Show that P(n,2) = 2^(n-1) - 1

There is no bijection between any set A and its power set P(A) of A.

There is no bijection between any set A and its power set P(A) of A. For finite sets, proof is trivial since |A| = n and |P(A)| = 2^n. For finite sets, this is done by contradiction. Suppose there is a bijection $ between a set A and its power set P(A). Consider the set B={x|x is a member A where x is not a member $(x)}For e ...continues

Discrete. Send answer as attachment

Please see the attached file for full problem description. The double bracket notation is pronounced " n multichoose k". The doubled parentheses remind us that we may include elements more than once.

Discrete.

Please see the attached file for full problem description.

Rabbit Farm

You own a rabbit far. Every week each pair of rabbits has two baby rabbits. However, the baby rabbits first reproduce when they are two weeks old. Initially you have a pair of newborn rabbits. How many pairs you have after n weeks?

Hogatt's Theorem

Prove Hogatt's Theorem: Any integer number can be written as a sum of terms of Fibonacci series. See attached file for full problem description.

Discrete Structures (Recursively Defined Sequences; Basis Definitions; Induction)

The attached sequences are "recursively defined" (for definiton please see attachment). Please answer the questions regarding each sequence. Thank you.

Recursive Definition Sequence

The attached sequence of numbers is recursively defined. Please complete the sequence.