Properties of Functions - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CS 173:

Discrete Mathematical Structures

Functions - misc. properties

f(A ∩ B) ⊆ f(A) ∩ f(B)?

f(A ∩ B) = {x : ∃a ∈ (A ∩ B), f(a) = x}

Choose an arbitrary b ∈ f(A ∩ B), and show that it must also be an element of f(A) ∩ f(B).

So, ∃a ∈ (A ∩ B) such that f(a) = b.

If a ∈ A (it is), then f(a) = b ∈ f(A).

If a ∈ B (it is), then f(a) = b ∈ f(B).

b ∈ f(A), and b ∈ f(B), so b ∈ f(A) ∩ f(B).

CS 173

Functions - injection

A function f: A → B is one-to-one

(injective, an injection) if ∀a,b,c,

(f(a) = b ∧ f(c) = b) → a = c

Not one-to-one

Every b ∈ B has at most 1 preimage.

Michael Tito Janet Cindy Bobby

Katherine Scruse

Carol Brady

Mother Teresa

CS 173

Functions - surjection

A function f: A → B is onto (surjective,

a surjection) if ∀b ∈ B, ∃a ∈ A f(a) =

b

Not onto

Every b ∈ B has at least 1 preimage.

Michael Tito Janet Cindy Bobby

Katherine Scruse

Carol Brady

Mother Teresa

Functions - examples

Suppose f: R +^ → R +^ , f(x) = x 2.

Is f one-to-one?

Is f onto?

Is f bijective?

yes yes yes

Functions - examples

Suppose f: R → R +^ , f(x) = x 2.

Is f one-to-one?

Is f onto?

Is f bijective?

no yes no

Functions - composition

Let f:A→B, and g:B→C be functions.

Then the composition of f and g is:

(g o f)(x) = g(f(x))

Functions - a little problem

Let f:A→B, and g:B→C be functions.

Prove that if f and g are one to one, then g o f :A→C is one to one.

Recall defn of one to one: f:A->B is 1to1 if f(a)=b and f(c)=b --> a=c.

Suppose g(f(x)) = y and g(f(w)) = y. Show that x=w.

f(x) = f(w) since g is 1 to 1.

Then x = w since f is 1 to 1.

Familiar functions

Polynomials: f(x) = a 0 x n^ + a 1 x n-1^ + … + an-1 x 1 + anx 0

Ex: f(x) = x 3 - 2x 2 + 15

Exponentials: f(x) = c dx

Ex: f(x) = 3 10x^ , f(x) = ex

Logarithms: log 2 x = y, where 2 y^ = x. In this course, log 2 n is written lg n. If we write log n, assume log 2 n.

Familiar functions

Ceiling: f(x) = x the least integer y so that x ≤ y.

Ex: 1.2 = 2; -1.2 = -1;  1  = 1

Floor: f(x) = x the greatest integer y so that x ≥ y.

Ex: 1.8 = 1; -1.8 = -2; -5 = -

Quiz: what is -1.2 + 1.1?

0

Two example algorithms.

I have a number between 0 and 63.

You ask a question, I’ll tell you yes

or no.

How long will it take you to find my

secret number?

Suppose it takes 1 time unit to answer a query about my number.

We really have no clue how long the algorithm takes to run, but we have an inkling that it depends logarithmically on n.

In this case the algorithm takes about lg n time units.

We say the algorithm has “order log n” running time.

Who wins the race?

The following graph gives times for

completing races of length x, for 4

different competitors.

Who is the tortoise?

time

distance

Who is the hare?

How would you describe blue’s performance?

At each distance, who wins?

Quiz time…

Describe this function:

Very slow growing.

Logarithmic function

Quiz time…

Describe this function:

Very fast growing.

Exponential function