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Sets and Logic

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.

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Sets and Logic

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  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 • [a,∞) = {x | a≤x} • (a, ∞) = {x | a<x} • (-∞,a] = {x | x≤a} • (-∞,a) = {x | x<a} • The whole real line R = (-∞, ∞) 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 A B

  9. 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

  10. 1 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 0 1 2 -1 2

  11. 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

  12. 3 4 -1 0 1 2 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 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

  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 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!

  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 • x2 = 4 ⇔ x = 2 or x = -2 • a2 + b2 = 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

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