Functions


















Example: grades

Consider a set of students in a class.
Suppose each student is assigned a grade.
The assignment can be represented as a set of pairs.

	students = { Jim, Ann, Bob, Sue, Joe }
	grades = { A, B, C, D, E }

	assignment = { (Jim, B), (Ann, A), (Bob, C), (Sue, B), (Joe, A) }

Is this set of pairs a relation?

Is any student assigned more than one grade?

Is more than one student assigned the same grade?

Is this set of pairs a function?


















What's a Function?

	A function from A to B is an assignment
	of an element from B to each element of A.

	An element of A can't be mapped to more than one element of B.

What's a Total Function?

	Every element of A has exactly one element from B assigned to it.

What's a Partial Function?

	Some elements of A have no assignment.


















Classwork
You may work with a partner.

Given domain A and codomain B, classify each relation
as a total function, a partial function, or not a function.

	A = { 1, 2, 3, 4 }
	B = { a, b, c }

	F1 = { (1,a), (2,b), (3,b) }

	F2 = { (2,a), (2,b), (2,c) }

	F3 = { (1,b), (2,b), (3,a), (4,a) }

	F4 = { (1,b), (2,a), (3,c), (4,a), (2,b) }


















Properties of Functions

What's an Injection (one-to-one)?

	A function from A to B where each member of B is used at most once.
	No member of B is assigned to more than one domain element.

What's a Surjection (onto)?

	A function from A to B where each member of B is used at least once.
	Every member of B is assigned to some domain element.

What's a Bijection (one-to-one correspondence)?

	A total function that is both one-to-one and onto.


















Classwork
You may work with a partner.

Given domain A and codomain B, classify each function
as an injection, a surjection, a bijection, or none.

	A = { 1, 2, 3, 4 }
	B = { a, b, c, d }

	F5 = { (1,b), (2,b), (3,a), (4,c) }

	F6 = { (1,c), (2,b), (3,d), (4,a) }

	F7 = { (1,a), (2,c), (3,a), (4,d) }

	F8 = { (1,a), (2,b), (3,c) }