1. Terminology and Symbols
csci 432 Lecture one

2. 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

3. 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

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

5. 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

6. 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?
??????????????

7. 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 }

8. 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

9. 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

10. 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?

11. 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).

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

13. 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

14. 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) }

15. 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)

16. Practice L = A - ( B U C ) R = (A - B) ∩ (A - C) L = R True or Falseif
L = A - ( B U C )
and
R = (A - B) ∩ (A - C)
then
L = R

17. 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}