Terminology and Symbols csci 432 Lecture one

Sets a Set is a collection of objects L = { a, b, c, d } a is an element or member of L a ∈ L z ∉ L {3, 1, 9} = {1, 3, 9} {7} is a singleton

Sets … A = { a, b, c, d } B = { a, b } C = { e, f, g, … z } B ⊆ A B is a subset of A ∅ ⊆ A the empty set is a subset of A C ⊄ A C is not a subset of A B ⊂ A B is proper subset of A, A ≠ B

Check your understanding… A = { a, b, … z} B = { a, b, c } Which are True? A ⊆ B X B ⊆ A ✔ B ⊆ B ✔ B ⊂ A ✔

Sets … A = { 1, 3, 9 } B = { 3, 5, 7 } A∩B = { 3 } intersection A∪B = { 1, 3, 5, 7, 9 } union A - B = { 1, 9 } difference

Closed Form A ∪ B = { x : x ∈ A or x ∈ B } A ∩ B = { x : x ∈ A and x ∈ B } A - B = { x : x ∈ A and x ∉ B } ?????????????? "Closed Form" is a way to mathematically denote something. The first line denotes Union. What is the closed form for intersection and difference? ??????????????

Practice N = {0, 1, 2, 3, … } what is { 2n : n ∈ N } Write the definition of positive odd numbers. O = { x : x ∈ N and x is not divisible by 2 } So what about this? O = { x : 2x-1 and x ∈ N }

Strings alphabet = set of symbols string = finite sequence from the alphabet language = set of words |w| = length of a word w wR = reverse of word w So… What is this called? w = wR

Ordered Pairs (a, b) a and b are components of the ordered pair (a, b) ≠ (b, a) order matters (a, b) ≠ { a, b } ordered pair v. set

Cartesian Product A = { 1, 2 } B = { a, b, c } A x B = { (1,a), (1,b), (1,c), (2,a), (2,b), (2,c) } So… Does A x B = B x A?

Functions f : R -> S f is a function that takes input from set R and returns elements from set S. f ⊆ R x S and for each x ∈ R there is at most one y ∈ S such that f(x) = y or (x, y).

One to One Functions f (x) = y is one to one if f -1 (y) = x

Practice True or False {2, 4, 5} ⊆ { 2n : n ∈ N } What is |w| and wR for this word? ysaEooTsIsihT Let A = {1, 2, … 100} and B = {a, b, c, d} How large is A x B? textbook page 9

Practice Suppose, R = {1, 2} and S = {a, b} Which is correct? RxS = { (1,a), (1,b), (2,a), (2,b) } RxS = ( {1,a}, {1,b}, {2,a}, {2,b} ) RxS = { (a,1), (a,2), (b,1), (b,2) } RxS = { (1,a), (2,a), (1,b), (2,b) }

Practice A ∪ A = A A ∩ A = A A ∪ B = B ∪ A ( A ∪ B ) ∩ A = A True or False A ∪ A = A A ∩ A = A A ∪ B = B ∪ A ( A ∪ B ) ∩ A = A ( A ∪ B ) ∩ C = (A ∩ C) ∪ (B ∩ C)

Practice L = A - ( B U C ) R = (A - B) ∩ (A - C) L = R True or False if L = A - ( B U C ) and R = (A - B) ∩ (A - C) then L = R

Practice ∅ ⊆ { ∅ } ∅ ∈ { ∅ } {a,b} ∈ {a, b, c, d} True or False ∅ ⊆ { ∅ } ∅ ∈ { ∅ } {a,b} ∈ {a, b, c, d} {a,b} ∈ {a, b, c, d, {a,b}} {a,b} ⊆ {a, b, c, d} What is ({1,3,5}∪{3,1})∩{3,5,7}