CSE 2500

Introduction to Discrete Systems

A) Variables
B) Math Statement
C) Set Subset
D) Cartesian Product
E) Relations and Functions

Examples:

1) If a number is greater than 2, then its square is greater than 4.
=> If x > 2, then x^2 > 4

2) Are those two numbers with the property that the sum of their squares is equal to the square of their sum.
=> a^2 + b^2 = (a+b)^2

*A universal statement says that a certain property holds true for all elements in a set.
* A conditional statement says if one thing is true, then some other thing has to be true.
* An essential statement says that there is at least one element for which the property is true.

Notation:
Upside down “A” means “For All”
Backwards “E” means “There Exists”
“V” is for “And”
“^” (Upside down V) means “Or”
“E” means for “Element”
“<->” means “Bi-conditional”

Ex:
1) There is a positive integer that is less than to every positive integer.
=> (backwards E) a positive integer n, such that (upside A) m E N, n < m.

2) “x E s” means s is an element of s
2b) “x e(slashed through it) s” means that x is not an element of s

Sets: [The order in which the elements of a set are listed is not important]
S = {1,2,3} , N = {1,2,3,4,5,….}

Subset:
Let A and B be two sets. A is a subset of B. A is less than or equal to B, if and only if, every element of A is also in B.
=> A lessthanequalto B <-> (upside A) x, if x E A, then x E B.

Examples of Elements in a Set:

*A = {1,2,3}
*1 E {1,2,3} => True

*{1} E {1,2,3} => False [Since this would be looking for a 1 and once again within a set]

Cartesian Product:
Given Elements a and b.(a,b) the ordered pair consists of a and b, with a as the 1st element pair and b as the second element.

*Ex1: (2,3) = (sqrt(4), 2*1.5)

*Ex2: A = {1,2,3}, B{a,b}
=> (1,a),(2,a),(3,b) and so on…