How many onto functions are possible

Discrete Structures

Readings Check, section 5.1

Read Section 5.1, pages 248 ( 252.

1) What is the notation A ( B called? What set does it produce?

2) Looking at example 5.3, what is |A ( B ( B ( B| ?

3) What is a binary relation from A to B?

4) Looking at example 5.5, and supposing that |C| = 4 and |D| = 5 how many possible relations are there from C to D?

5) In example 5.8 is (5, 28) in the relation R ?

6) Using the theorem on p. 252, how can (D ( F) ( (E ( F) be rewritten?

Discrete Structures

Readings Check, section 5.2

Read Section 5.2, pages 252 ( 258.

1) What is a function and how does it differ from a relation?

2) In example 5.9 why are R1 and R2 not functions?

3) Reading definition 5.4, how does the codomain of f differ from the range of f ?

4) What is the value of (4.25( ? What is the value of (4.25( ?

5) Suppose that |A| = 5 and |B| = 10. How many functions can possibly be made from A to B ? [see p. 255]

6) When is a function f : A ( B called one(to(one?

7) Again if |A| = 5 and |B| = 10, what is the number of one(to(one functions from A to B ?

Discrete Structures

Readings Check, section 5.3

Read Section 5.3, pages 260 ( 265.

1) Why is the case of example 5.20 not an onto function?

2) Does an onto function also have to be one(to(one? If so then explain why. If not then give an example of an onto function that is not one(to(one..

3) Does a one(to(one function also have to be onto? If so then explain why. If not then give an example of a one(to(one function that is not onto.

4) Looking at example 5.23, if A = {x, y} then how many onto functions are there from A onto B? In other words, answer the question at the end of the example when m = 2. Explain without reference to the formula why the answer comes out like that.

5) The blue box at the top of page 262 gives a formula for the number of onto functions from domain A onto range B. With |A| = m and |B| = n, what does this formula equal in terms of distributions of objects into containers? [be precise]

6) What does the formula in the box on page 263 calculate? [That is, the formula for Stirling numbers of the second kind.]

Weekly Summary, Week 4, Chapter 5, Discrete Structures Name:

Due on Sunday, February 12, by midnight.

1) Give a recursive definition (similar to example 5.8) for the relation R containing the ordered pairs:

(0, 2), (2, 4), (4, 16), (6, 256), …

2) Let A = {1, 2, 3} and B = {1, 2, 3, 4, 5}. How many one(to(one functions f : A ( B satisfy f(1) = 1 ?

3) Is it true or false that (A ( B) ( (A ( C) = A ( (B ( C) for any sets A, B, C ? [Remember that for a statement to be considered true, it must be true in all cases.]

4) Suppose that |A| = |B| = n.

a) If n = 1 how many onto functions f : A ( B are possible?

b) If n = 2 how many onto functions f : A ( B are possible?

c) If n = 3 how many onto functions f : A ( B are possible?

d) In general, for |A| = |B| = n, how many onto functions f : A ( B are possible?

[Hint: there is an easy pattern. Do the case of n = 4 if you still do not see it.]

5) Consider the number 510,510 which factors into prime factors as 2(3)(5)(7)(11)(13)(17). In how many ways can 510,510 be factored into 3 factors, all greater than 1?

[Note: this is an application of the formula for Stirling numbers of the second kind, as per example 5.28.

To avoid doing the actual calculation see the table at: https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind