SWER ENG 2DM3 Tutorial 2

Transcription

1 Tutorial 2 Min Jing Liu Department of Computing and Software McMaster University Sept 22, 2011 Tutorial 2

2 Outline 1 Set 2 Function 3 Sequences and Summation

3 Set Set Definition: A set is an unordered collection of objects.

4 Set Set Definition: A set is an unordered collection of objects. Set Representation Define set by directly listing all its elements (explicit), E.g: A = {1, 3, 5, 7, 9} A = {{1, 1}, {3, 2}, {5, 2}, {7, 1}} Define set by using a rule or semantic description (implicit), E.g: A is the set of colors of the Canada flag. A = {x x is a prime number and x < 1000}.

5 Set Set Representation Empty Set: Let S be a set, and a be an element. a S : a is an element of set S a / S : a is not an element of set S. Subset: A B x[x A x B] Set Equality: A = B (A B) (B A) Proper Subset: A B (A B) (A B)

6 Set Power Set Power Set Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S). : What is the power set of {0, 1, 2}? Solution: The power set P{0, 1, 2} is the set of all subset of {0, 1, 2}. Hence, P({0, 1, 2}) = {, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}. If a set has n elements, then its power set has 2 n elements.

7 Set Set Operations Union: A B = {x x A x B} Intersection: A B = {x x A x B} Difference: A B = {x x A x / B} Complement: let U be the universal. Ā = U A.

8 Which of the followings is true? If A B =, then A = B (A B) B = A B (A B) (A C) = A (B C)

9 Function Function Let A and B be sets. If f is a function from A to B, we have f : A B. Let f 1 and f 2 be functions for A to R, then f 1 + f 2 and f 1 f 2 are also functions from A to R defined by (f 1 + f 2 )(x) = f 1 (x) + f 2 (x) (f 1 f 2 )(x) = f 1 (x)f 2 (x)

10 Function : Let f 1 and f 2 be functions from R to R such that f 1 (x) = 3x 2 and f 2 (x) = x 2x 2. What are the function (f 1 + f 2 )(x) and (f 1 f 2 )(x)? Solution: (f 1 + f 2 )(x) = f 1 (x) + f 2 (x) = 3x 2 + x 2x 2 = x + x 2 (f 1 f 2 )(x) = 3x 2 (x 2x 2 ) = 3x 3 6x 4

11 Function Function One-to-one Onto

12 Function Inverse Function Let f be a one-to-one correspondence from the set A to the set B. The inverse function of f is denoted be f 1 : Hence, f 1 (b) = a when f (a) = b.

13 Function Composite Function Let g be a function from the set A to the set B and let f be a function from the set B to the set C. The composition of the function f and g, denoted by f g: In general: g(f (x)) f (g(x)) (f g)(a) = f (g(x))

14 : f (x) = x and g(x) = 2 x, find f g and g f when x = 1 Solution: f g = f (g(x)) = f (2 x) = (2 x) = x 2 4x + 7 (f g)(1) = = 4. g f = g(f (x)) = g(x 2 + 3) = 2 (x 2 + 3) = x 2 1 (g f )(1) = 1 1 = 2.

15 : Find the inverse function g 1 (x) to g(x) = 5x 2 2,x 1 Solution: Let y = g(x) Thus g 1 (x) = x+2 5 y = 5x 2 2 (1) y + 2 = 5x 2 (2) 1 5 (y + 2) = x 2 (3) y + 2 = x (4) 5

16 Sequence and Summation Sequence Definition A sequence is a discrete structure used to represent an ordered list {a n } Each a n is called the n th term of the sequence.

17 Sequence and Summation Sequence Consider the sequence The terms are: a 1 = (1 + 1) 1 = a 2 = ( )2 = a 3 = ( )3 = a 4 = ( )4 = a 5 = ( )5 = {(1 + 1 n )n } n=1

18 Sequence and Summation Summation n j=m a j = a m + a m a n 1 + a n j is the index of summation, m is the lower limit, and n is the upper limit. Sometimes, it is useful to change the lower/upper limits, which can be done in a straightforward manner (though we must be careful). n j=1 a j = n 1 j=0 a j+1

19 : What is the value of n i=1 i2i? Solution: