Download presentation

1
**More Functions and Sets**

Rosen 1.8

2
Inverse Image Let f be an invertible function from set A to set B. Let S be a subset of B. We define the inverse image of S to be the subset of A containing all pre-images of all elements of S. f-1(S) = {aA | f(a) S} S A f(a) a B

3
**Let f be an invertible function from A to B. Let S be a subset of B**

Let f be an invertible function from A to B. Let S be a subset of B. Show that f-1(S) = f-1(S) What do we know? f must be 1-to-1 and onto S A B b1 a2 b2 a1 f -1 f-1(S)

4
**Let f be an invertible function from A to B. Let S be a subset of B**

Let f be an invertible function from A to B. Let S be a subset of B. Show that f-1(S) = f-1(S) Proof: We must show that f-1(S) f-1(S) and that f-1(S) f-1(S) . Let x f-1(S). Then xA and f(x) S. Since f(x) S, x f-1(S). Therefore x f-1(S). Now let x f-1(S). Then x f-1(S) which implies that f(x) S. Therefore f(x) S and x f-1(S) If x f-1(S). Then f(x) S after taking the inverse of both sides (i.e., x = f-1(y) means y = f(x); therefore y S).

5
**Let f be an invertible function from A to B. Let S be a subset of B**

Let f be an invertible function from A to B. Let S be a subset of B. Show that f-1(S) = f-1(S) Proof: f-1(S) = {xA | f(x) S} Set builder notation = {xA | f(x) S} Def of Complement = f-1(S) Def of Complement

6
**Floor and Ceiling Functions**

The floor function assigns to the real number x the largest integer that is less than or equal to x. x x = n iff n x < n+1, nZ x = n iff x-1 < n x, nZ The ceiling function assigns to the real number x the smallest integer that is greater than or equal to x. x x = n iff n-1 < x n, nZ x = n iff x n < x+1, nZ

7
**Examples 0.5 = 1 0.5 = 0 -0.3 = 0 -0.3 = -1 6 = 6 6 = 6**

-3.4 = -3 3.9 = 3

8
**Prove that x+m = x + m when m is an integer.**

Proof: Assume that x = n, nZ. Therefore n x < n+1. Next we add m to each term in the inequality to get n+m x+m < n+m+1. Therefore x+m = n+m = x + m x = n iff n x < n+1, nZ

9
**Let xR. Show that 2x = x + x+1/2**

Proof: Let nZ such that x = n. Therefore n x < n+1. We will look at the two cases: x n + 1/2 and x < n + 1/2. Case 1: x n + 1/2 Then 2n+1 2x < 2n+2, so 2x = 2n+1 Also n+1 x + 1/2 < n+2, so x + 1/2 = n+1 2x = 2n+1 = n + n+1 = x + x+1/2 n n+1

10
**Let xR. Show that 2x = x + x+1/2**

Case 2: x < n + 1/2 Then 2n 2x < 2n+1, so 2x = 2n Also n x + 1/2 < n+1, so x + 1/2 = n 2x = 2n = n + n = x + x+1/2

11
**Characteristic Function**

Let S be a subset of a universal set U. The characteristic function fS of S is the function from U to {0,1}such that fS(x) = 1 if xS and fS(x) = 0 if xS. Example: Let U = Z and S = {2,4,6,8}. fS(4) = 1 fS(10) = 0

12
**Let A and B be sets. Show that for all x, fAB(x) = fA(x)fB(x)**

Proof: fAB(x) must equal either 0 or 1. Suppose that fAB(x) = 1. Then x must be in the intersection of A and B. Since x AB, then xA and xB. Since xA, fA(x)=1 and since xB fB(x) = 1. Therefore fAB = fA(x)fB(x) = 1. If fAB(x) = 0. Then x AB. Since x is not in the intersection of A and B, either xA or xB or x is not in either A or B. If xA, then fA(x)=0. If xB, then fB(x) = 0. In either case fAB = fA(x)fB(x) = 0.

13
**Let A and B be sets. Show that for all x, fAB(x) = fA(x) + fB(x) - fA(x)fB(x)**

Proof: fAB(x) must equal either 0 or 1. Suppose that fAB(x) = 1. Then xA or xB or x is in both A and B. If x is in one set but not the other, then fA(x) + fB(x) - fA(x)fB(x) = 1+0+(1)(0) = 1. If x is in both A and B, then fA(x) + fB(x) - fA(x)fB(x) = 1+1 – (1)(1) = 1. If fAB(x) = 0. Then xA and xB. Then fA(x) + fB(x) - fA(x)fB(x) = – (0)(0) = 0.

14
**A B AB fAB(x) fA(x) + fB(x) - fA(x)fB(x)**

Let A and B be sets. Show that for all x, fAB(x) = fA(x) + fB(x) - fA(x)fB(x) A B AB fAB(x) fA(x) + fB(x) - fA(x)fB(x) (1)(1) = 1 (1)(0) = 1 (0)(1) = 1 )-(0)(0) = 0

Similar presentations

OK

Functions (Mappings). Definitions A function (or mapping) from a set A to a set B is a rule that assigns to each element a of A exactly one element.

Functions (Mappings). Definitions A function (or mapping) from a set A to a set B is a rule that assigns to each element a of A exactly one element.

© 2018 SlidePlayer.com Inc.

All rights reserved.

By using this website, you agree with our use of **cookies** to functioning of the site. More info in our Privacy Policy and Google Privacy & Terms.

Ads by Google

Ppt on my school days Ppt on 2nd world war movies Ppt on standing order mandate Ppt on mauryan empire Ppt on stock market in india Free ppt on save environment Ppt on building information modeling bim Ppt on central limit theorem equation Ppt on stages of economic development Ppt on solid dielectrics inc