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

Prove that 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 each element x A, since $ is a function from A to the power set of A, &(x) ia a subset of A. By our earlier assumption on set theory that every element is either in a set or ot in a set, for each x A, w can certainly ask if x is a member of $(x). Therefore, the set B is well definedB. We know B is a subset of A and $ is a bijection from A to its power set P(A). For... click for more

Discrete Math : N multichoose K Proof

Please see the attached file for the 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?