Friday, Sept 14 Today we will probably finish Course Notes 2.3: Predicates and Quantifiers. After that, we might begin Course Notes 1.1: Definitions in Set Mathematics.
EXAMPLE Let P(x): 'x is a JRR Tolkien authority;' Q(x): 'x is a nerd;' R(x,y): 'x has read y.' The universe for x is the set of all people, the universe for y is the set of all JRR Tolkien novels. True/false The symbolization for 'Every nerd who has read at least one JRR Tolkien novel is a JRR Tolkien authority' is x[(q(x) yr(x, y)) P(x)] A. True B. False
EXERCISE Select the negation of Every wolverine is bitey. A. No wolverine is bitey. B. At least one wolverine is bitey. C. At least one wolverine is not bitey. D. None of these.
Negating Quantified propositions (DeMorgan s Laws for Quantifiers) 1. xp(x) x P(x) 2. xp(x) x P(x) That is, the negation of P(x) is true for all x is There is at least one x for which P(x) is not true. That is, the negation of There is at least one x for which P(x) is true is P(x) is false for all x which can also be stated There is no x for which P(x) is true.
EXAMPLE Write the negation of At least one badger is cuddly. Answer
EXAMPLE What is the negation of x[(q(x) R(x, The Hobbit)) P(x)]? A. x[ (Q(x) R(x, The Hobbit)) P(x)] B. x[ (Q(x) R(x, The Hobbit)) P(x)] C. x[(q(x) R(x, The Hobbit)) P(x)] D. x[(q(x) R(x, The Hobbit)) P(x)] E. None of these In words, the negation of Every nerd who has read The Hobbit is a JRR Tolkien authority is
DeMorgan s Laws for Nested Quantifiers If there is more than one quantifier, then the negation operator should be passed from left to right across one quantifier at a time, using the appropriate DeMorgan Law at each step. EXAMPLE Use DeMorgan s Laws to negate: x y[q(x) R(x,y)]
SOMETHING ELSE Suppose U = {x 1, x 2 }, P(x), Q(x) are predicates, and P(x 1 ) is T P(x 2 ) is F Q(x 1 ) is F Q(x 2 ) is T Evaluate each of these quantified propositions: 1. x[p(x) Q(x)] 2. xp(x) xq(x) Which propositions are true? A. Prop. 1 only is true B. Prop. 2 only is true C. Prop. 1 and Prop. 2 are both true D. Both are false
The result of the previous exercise proves the following fact (from the end of Chapter 2 Section 3) about distributing quantifiers. For any propositions P(x), Q(x) x[p(x) Q(x)] xp(x) xq(x) That is, the universal quantifier does/does not distribute over a disjunction.
Referring to the previous example, note that x[p(x) Q(x)] is. xp(x) xq(x) is. This proves: For any predicates P(x), Q(x) x[p(x) Q(x)] xp(x) xq(x) That is, the existential quantifier does/does not distribute over a conjunction. Finally, two other facts that cannot be proven with just a single example: x[p(x) Q(x)] xp(x) xq(x) x[p(x) Q(x)] xp(x) xq(x)
Course Notes 1.1: Definitions in set mathematics While introducing material from Course Notes 2.3, we also introduced, as needed, some terminology from Course Notes 1.1. Before moving into Chapter 3, we will finish Course Notes 1.1. The terminology and notation of set mathematics is common vocabulary for almost any theoretical or higher level discussion in math and computing.
Chapter 1, Section 1: Set Mathematics, definitions Set A set S is an unordered collection of objects. Element The objects contained in a set are called its elements or members. If x is an element of set S, we write If x is not an element of set S, we write Universe In each situation there is an underlying universal set U, either stated or implied, from which elements may be drawn.
Methods for defining or describing particular sets Roster method The elements of a set are listed, separated by commas, within curly braces. Example S = {a, b, c, d} When we are using the roster method, the order in which the elements are listed is not important, and redundant listing of elements is ignored. S = {a, b, c, d} = { } = { } T/F (a, b, c, d) = {a, b, c, d}
Roster method with ellipses Suppose our universe is the set of integers. Within this universe we will define a set T as follows: T = { 1, 0, 1,..., 10} When we use ellipses like this, the assumption is that the context makes their meaning clear. A set is well defined if there is no uncertainty as to whether or not a particular object belongs in it. A set that is not well defined is said to be ill defined.
Set-builder notation defines a set by describing, rather than listing, its elements. Let the universe be the set of real numbers. Within this universe define a set V as follows: V = {x x > 1.3 and x < 2π} This is read V is the set of all x such that x is greater than 1.3 and x is less than 2π. We should recognize that V is the infinite set containing all real numbers between 1.3 and 2π. In other words, V is the open interval. The vertical bar represents such that. We can also use a colon instead of the vertical bar: V = EXAMPLE Using set builder notation, the set T that was previously defined using the roster method and ellipses can be alternatively defined: