Lecture 10 21 127 Concepts of Math 09 19 2012 Administrivia Homework 3 is due tomorrow For Problem 6 you don t need to prove that addition is commutative The point of the question is to see whether you can interpret the sets appropriately The only algebraic issue comes up in part d For Problem 2 you shouldn t need to write out 12 cases Furthermore you should not be saying something about maximum or minimum values This is not a rigorous concept In fact any meaningful definition of max or min would be stated in terms of an inequality which is what you are trying to deduce anyway Just take an arbitrary element of A and write down something you know about such an element in terms of a few inequalities Manipulate those to conclude the element is in B as well Quantifiers Some examples of usage and equivalence of x R x2 0 For all real numbers x x2 0 Every real number x satisfies x2 0 Whenever x is a real number x2 0 Some examples of usage and equivalence of x R x2 4x 4 0 There exists a real number x such that x2 4x 4 0 There is a real number x that satisfies x2 4x 4 0 Important Observation The phrase such that comes always and only after an quantification Some bad examples x x 7 y Z w Z w z 2 z 2 4 2 t t N We always have to specify a set when we quantify Example Some more good examples x R x2 0 S P N 7 S a Z b Z a b 0 1 The Goldbach Conjecture Let X be the set of even natural numbers that are at least 4 Let P be the set of prime numbers Define Q n a b to be n a b n X a b P Q n a b n N x R n x x R y R y x y R x R y x x R n N n x 3 3 The order of quantification really matters Indexed Set Operation Definitions Using Quantifiers We can now redefine indexed set operations using these symbols Suppose I is an index set and i I Ai U for some universal set U Then Ai x U k I x Ak A x U i I x A i I i i i I Logical Negation Definition We use the symbol to mean logical negation That is if P is a mathematical statement then P is the statement with the opposite truth value of P Negating Claims n N n 12 This says It is False that there exists a natural number n that is less than 12 This is the same as saying Every natural number is at least 12 n N n 1 2 Negating Claims x R x2 0 This says It is False that every real number x satisfies x2 0 This is the same as saying There exists a real number x that satisfies x2 0 x R x2 0 We can put these two methods together and negate any chain of quantifiers n N x R n x is the same as n N x R n x 2 x R y R y x3 is the same as x R y R y 6 x3 Let P y R y 0 P P x y R 0 x3 Logical Connectives We need to know how to take quantified statements and variable propositions and combine them to make more intricate and interesting statements And Definition We use the symbol between two mathematical statements to mean and For instance we read P Q as P and Q This is referred to as the conjunction of P and Q The truth value of P Q is True when both P and Q are true and the truth value is False otherwise Examples 1 3 5 x R x 1 3 5 x Q x 1 3 4 x R x2 0 True 2 0 False 2 2 False Or In math we use the inclusive or not XOR Definition We use the symbol between two mathematical statements to mean or For instance we read P Q as P or Q This is referred to as the disjunction of P and Q The truth value of P Q is True when at least one of P and Q is true even when both are true and the truth value is False otherwise Examples 1 3 5 x R x 1 3 5 x R x 1 3 4 x R x2 0 True 2 0 True 2 0 False Remember and are set operations they only apply to sets Do not mix notation with and Conditional Statements This is the hardest logical connective to work with and continually gives students some problems so we want to be extra careful and clear about this one We want the statement P implies Q or If P then Q to have the truth value True when the truth of Q necessarily follows from the truth of P that is we want this statement to be True if whenever P is true Q is also true Since this is the hardest connective to suss out semantically let s introduce the idea of a truth table to make the notation easier P T T F F Q T F T F P F F T T P Q T F F F P Q T T T F P Q T F T T 3 P Q T F T T Q P T T F T P Q T F F T Definition We use the symbol between to mathematical statements to mean implies or If then For instance we read P Q as P implies Q or If P then Q This is referred to as a conditional statement The truth value of P Q is True when Q is also true whenver P is true the truth value is False only when P is true and yet Q is false We refer to P as the hypothesis of the conditional statement and Q as the conclusion That key word whenever in the definition should indicate to you why the false hypothesis case makes sense When we definitely know P is true and can deduce that Q is also necessarily true then we get to declare P Q as True If P wasn t true to begin with we cannot declare P Q to be false We only get to say P Q is false when Q did not necessarily follow from P that is when there is an instance where P is true but Q is false Here are some more examples to help you get the idea 1 3 5 x R x 1 3 4 x R x2 0 2 True 0 True 1 3 5 I am dead True 1 1 2 0 1 False 2 0 0 x R x 0 False Pythagorean Theorem 1 1 True 0 1 1 1 True Notice that the second to last implication is True …