surjection

Relations and Functions between Small Sets The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. A relation between two sets is a subset of their Cartesian product. The subset's pairs are plotted in the graph. Click the dots on the left to add or delete pairs from the relation. Contributed by: Chris Boucher

Math Ninja: Group Theory 5-3 Bijections, Surjection (onto), injection (one to one), examples, and main step in alternative proof that rationals are countable.

Functions 3 Introduction of injective and surjective functions, with definition at the end

Functions 5 More examples with more formal proofs of injectivity and surjectivity, and proofs that functions are not injective and surjective

N onto Q The formulas including inverse are located here: n is mapped onto (p,q) which is associated with the rational number p/q.

Math Ninja: Group Theory 5-1 Bijections, Surjection (onto), injection (one to one), examples, and main step in alternative proof that rationals are countable. omework (For more reading on this, try wikipedia, planet math, or www.stanford.edu "function slides): 1. Find a bijection between the natural numbers (include 0) and even integers. Also find a bijection between the natural numbers (include 0) and odd integers. Don't forget to prove they are bijections. 2. Prove that the function f(x)=x^2 is not injective on the reals. Also prove that it is not surjective. 3. Label the square of the dihedral group D8 with 1,2,3,4 clockwise with 1 in the upper left corner (like in the 4 corners game, denote this (1,2,3,4) ). This is the identity position in D8. But this can also be considered the identity in a specific some group of S4 (which we will calculate). Rotating once, we get (4,1,2,3). So 1 went to 2, 2 went to 3, 3 went to 4, and 4 went to 1. So this is the element (1 2 3 4) of S4. Rotating again, we get (3, 4, 1, 2). So now we have (1 3)(2 4) of S4. So r maps to (1 2 3 4) and r^2 maps to (1 3)(2 4). Work about the entire correspondence for D4 and prove it is a bijection. Moreover, prove that all the elements of S4 that the D4 elements map to form a subgroup. 4. Prove that any finite sequence of english letters (eg ABCDDDME) has a bijection with the natural numbers that are greater than or equal to 2 (hint, think the RSA presentation with Wesley, James, and Prannay. They used power of primes multiplied ...

Functions 4 Looking at some examples of functions to see if they are injective and surjective, introducing bijective functions at the end

Lectures 26 - Functions (Contd.) Discrete Mathematical Structures

Math Ninja: Group Theory 5-2 Bijections, Surjection (onto), injection (one to one), examples, and main step in alternative proof that rationals are countable.

Math Ninja: Group Theory 5-4 Bijections, Surjection (onto), injection (one to one), examples, and main step in alternative proof that rationals are countable.

Surjective onto and Injective one to one functions

Injective, Surjective, Bijective The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. In the function mapping x-gtf(x), the domain is all x values and the range is all f(x) values. Contributed by: Ed Pegg Jr

Surjective (onto) and Injective (one-to-one) functions Introduction to surjective and injective functions