1 / 10

Discrete Mathematics Math 6A

Discrete Mathematics Math 6A. Instructor: M. Welling. 1.1 Propositions. Logic allows consistent mathematical reasoning. Many applications in CS: construction and verification computer programs, circuit design, etc.

Download Presentation

Discrete Mathematics Math 6A

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Discrete MathematicsMath 6A Instructor: M. Welling

  2. 1.1 Propositions • Logic allows consistent mathematical reasoning. • Many applications in CS: construction and verification computer programs, • circuit design, etc. Proposition:A statement that is either true (T) or false (F). example: Toronto is the capital of Canada in 2003 (F). 1+1=2 (T). counter-example: I love this class. Compound Propositions:New propositions formed by existing propositions and logical operators. Let “P” be a proposition. Then (“NOT P”) is another one stating that: It is not the case that “P”.

  3. 1.1 Propositions truth table example: P: Today is Tuesday. NOT P: Today is not Tuesday. NOT is the negation operator. Another class of operators are the “connectives”. P AND Q conjunction P OR Q disjunction , inclusive OR. P XOR Q exclusive OR. example: Bob is married to Carol. (T) Bob is married to Betty or to Carol. (T) Bob is married to Betty and to Carol (F).

  4. 1.1 Implications Implication : P  Q , P IMPLIES Q. P is hypothesis, Q is consequence. some names: if P then Q, Q when P, Q follows from P, P only if Q. example: If you make no mistakes, then you’ll get an A. Bidirectional implication: PQ , P if and only if (iff) Q. Implications are often used in mathematical proofs. Consider: P  Q. converse: Q  P. contra-positive: (NOT Q)  (NOT P) (equiv.) inverse: (NOT P)  (NOT Q). weird?

  5. 1.1 Precedence, Bits. Order of precedence: NOT, AND, OR, XOR, , . example: PQ AND NOT R = P (Q AND (NOT R) ). Bits are units of information. 1=T, 0=F. Bit-strings are sequences of bits: 00011100101010 We can use our logic operators to manipulate these bit-strings: example: 0110 AND 1100 = 0100 puzzle: Is this a proposition: “This statement is false”? if S = T  S = F, if S = F  S = T whoa: it is neither true nor false!

  6. 1.2 Propositional Equivalences Tautology:Proposition that is always true. for example: P OR (NOT P). Contradiction:Proposition that is always false. for example:P AND (NOT P). Others: Contingencies. Two propositions are logically equivalent if P  Q is always true (tautology). This is denoted by . Example: Morgan’s Law:

  7. 1.2 Propositional Equivalences Proving equivalences by truth tables can easily become computationally demanding: equivalence with 1 prop.: truth table has columns of size 2. equivalence with 2 prop.: ..................................................4. equivalence with 3 prop.: ..................................................8. equivalence with n prop.: ................................................... (How many times do we need to fold the NY-times to fit between the earth and the moon ?) Solution: we use a list of known logical equivalences (building blocks) and manipulate the expression. See page 24 for a list of equivalences.

  8. 1.3 Predicates & Quantification Let’s consider statements with variables: x > 3. x is the subject. >3 is the predicate or property of the subject. We introduce a propositional function, P(x), that denotes >3. If X has is a specific number, the function becomes a proposition (T or F). example: P(2) = F, P(4) = T. More generally, we can have “functions” of more than one variable. For each input value it assigns either T or F. example: Q(x,y) = ( x=y+3 ). Q(1,2) = ( 1=2+3 ) = F Q(3,0) =( 3=0+3)=T

  9. 1.3 Predicates and Quantification We do not always have to insert specific values. We can make propositions for general values in a domain (or universe of discourse): This is called: quantification. Universal Quantification: P(x) is true for all values of x in the domain: Existential Quantification: There exists an element x in the domain such that P(x) is true: example: domain x is real numbers. P(x) is x > -1. (F : counter-example: x=-2) (T) example: domain is positive real numbers, P(x) is x>-1. (T) (T)

  10. 1.3 Binding & negations A variable is bound if it has a value or a quantifier is “acting” on it. A statement can only become a proposition if all variables are bound. example: x is bound, y is free. The scope of a quantifier is the part of the statement on which it is acting. example: scope x scope y We can also negate propositions with quantifiers. Two important equivalences: It is not the case that for all x P(x) is true = there must be an x for which P(x) is not true It is not true that there exists an x for which P(x) is true = P(x) must be false for all x

More Related