Sets and Logic including Set Notation and Venn Diagrams

Sets and Logics

1. Sets and Logic
Alex Karassev

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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!

8. 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

9. 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

10. 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

11. 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

12. -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 = ∅

13. 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

14. 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!

15. 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

16. 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