If is a set, the set is the set of all ordered pairs of elements of . Define a function by .

Is (0, 0) an element of the range of f?

State four elements of the range of f.

Determine if f is an injection, a surjection, both (a bijection), or neither. Prove your answers.

Let be defined by . Determine if g is an injection, a surjection, both, or neither. Prove your answers.

Let A = { 1, 2, 3, …, n } and B = { a, b, c }. If n = 1, how many functions are there from A to B (ie, with domain A and codomain B? If n = 2, how many are there? Find a formula for the number of functions from a set A of cardinality n to the set B (which has cardinality 3).

Let be the set of functions with domain { 1, 2, 3 } and codomain { 0, 1 }. That is, if . Find a bijection from to . That is, find a bijection whose domain is the set of functions from {1, 2, 3} to {0, 1} and whose codomainÂ is the set of subsets of {1, 2, 3}.

Suppose our universe is and P(x, y) is defined as x + 1 = y. Determine which of the following statements are true and explain why: