Sets and Logic

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Sets and Logic

• Alex Karassev

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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

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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 1ltxlt2
• Reads as A is the set of all x such that x is
  irrational and 1ltxlt2

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Interval notations

• Closed interval a,b is the set of all numbers
  not smaller than a and not bigger than b
• a,b x axb
• Example
• -1,3

x
-1
3
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Interval notations

• Open intervals (a,b) is the set of all numbers
  bigger than a and smaller than b
• (a,b) x altxltb
• Example
• (-1,3)

x
-1
3
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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 altxb
• Example
• (-1,3
• The interval a,b) is defined similarly

x
-1
3
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Infinite intervals
a

• a,8) x ax
• (a, 8) x altx
• (-8,a x xa
• (-8,a) x xlta
• The whole real line R (-8, 8)

a
a
a
Note 8 is not a number!
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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 B1,3,5 then B?A
• If A 0,10 and C -1,3), C is not a subset
  of A

A
B
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Union

• The union of two setsA and Bis the set of
  allelements x such thatx is in A OR x is in B
• NotationA ? B x x ? A or x ? B

A
B
A ? B
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Union

• Examples
• If A (-1,1) and B0,2then A ? B (-1,2
• If A (- 8,1 and B (1, 8)then A ? B (- 8,
  8) R

-1
1
0
2
-1
2
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Intersection

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

A
B
A n B
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Intersection

• Examples
• If A (-1,1) ? 2, 4 and B(0,3then A n B
  (0,1) ? 2, 3
• If A (- 8,1 and B (1, 8)then A n B empty
  set Ø

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Logic implications

• P? Q
• reads P implies Q or if P then Q
• Example a (true) statement All cats need food
  can be stated asx 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 !

Q
P
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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 trueit does not mean that Q ? P
  is true
• Example
• All cats need food is true, sox 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!

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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

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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