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-1 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-1 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-1 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

BA

Union

The union of two setsA and Bis the set of allelements x such thatx 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

-1 10 2-1 2

1

Intersection

The intersectionof two setsA and Bis the set of allelements x such thatx is in A AND x is in B

Notation:A ∩ B = { x | x ∊ A and x ∊ B}

A B

A ∩ B

-1 10 2 3 4

Intersection Examples

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, x2 = 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 x2 = 4 ⇔ x = 2 or x = -2

a2 + b2 = 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