Sets and Logic Alex Karassev

Elements of a set  a ∊ A means that element a is in the set A  Example: A = the set of all odd integers bigger than 2 but less than or equal to 11 3 ∊ A 4 ∉ A 15 ∉ A

Set builder notation  Example: A = the set of all odd integers bigger than 2 but less than or equal to 11 A = {3, 5, 7, 9, 11}  Example: A = the set of all irrational numbers between 1 and 2 A = {x| x is irrational and 1<x<2} Reads as A is the set of all x such that x is irrational and 1<x<2

Interval notations  Closed interval: [a,b] is the set of all numbers not smaller than a and not bigger than b [a,b] = {x | a≤x≤b}  Example: [-1,3] x 3

Interval notations  Open intervals: (a,b) is the set of all numbers bigger than a and smaller than b (a,b) = {x | a<x<b}  Example: (-1,3) x 3

Interval notations  Half-Open (half-closed) intervals: (a,b] is the set of all numbers bigger than a and smaller than or equal to b (a,b] = {x | a<x≤b}  Example: (-1,3]  The interval [a,b) is defined similarly x 3

Infinite intervals  [a,∞) = {x | a≤x}  (a, ∞) = {x | a<x}  (-∞,a] = {x | x≤a}  (-∞,a) = {x | x<a}  The whole real line R = (-∞, ∞) a a a a Note: ∞ is not a number!

Subsets  Set B is called a subset of the set A if any element of B is also an element of A  B ⊂A  Example If A = [0,10] and B={1,3,5} then B ⊂A If A = [0,10] and C = [-1,3), C is not a subset of A B A

Union  The union of two sets A and B is the set of all elements x such that x is in A OR x is in B  Notation: A ∪ B = { x | x ∊ A or x ∊ B} A B A ∪ B

Union  Examples If A = (-1,1) and B=[0,2] then A ∪ B = (-1,2] If A = (- ∞,1] and B= (1, ∞) then A ∪ B = (- ∞, ∞) = R

Intersection  The intersection of two sets A and B is the set of all elements x such that x is in A AND x is in B  Notation: A ∩ B = { x | x ∊ A and x ∊ B} A B A ∩ B

Intersection  Example s If A = (-1,1) ∪ [2, 4] and B=(0,3] then A ∩ B = ( 0, 1) ∪ [2, 3] If A = (- ∞,1] and B= (1, ∞) then A ∩ B = empty set = ∅

Logic: implications  P⇒ Q reads: “P implies Q” or if “P then Q” Example: a (true) statement “All cats need food” can be stated as x is a cat ⇒ x needs food  Implications can be true or false. For example, x 2 = x ⇒ x = 1 is false  “⇒” is not the same as “=” ! P Q

Logic: converse  A converse of P ⇒ Q is Q ⇒ P  Warning: if a statement is true it does not mean that its converse is true  i.e. if P ⇒ Q is true it does not mean that Q ⇒ P is true  Example: “All cats need food” is true, so x is a cat ⇒ x needs food is true x needs food ⇒ x is a cat (if x needs food then x is a cat) is false!

Logic: equivalence  Two statements P and Q are called equivalent if both implications P ⇒ Q and Q ⇒ P hold  Notation: Q ⇔ P (reads “Q is equivalent to P” or “Q if and only if P”)  Examples x 2 = 4 ⇔ x = 2 or x = -2 a 2 + b 2 = 0 ⇔ a = b = 0 A triangle is equilateral ⇔ All its angles are equal

Logic: negation  Notation: NOT P, also ⌉ P and P  Negation and implication P ⇒ Q is true if and only if NOT Q ⇒ NOT P is true!  Example: x is a cat ⇒ x needs food NOT (x needs food) ⇒ NOT (x is a cat) x does not need food ⇒ x is not a cat