Propositions - Discrete Mathematics and its Applications - Lecture Slides, Slides of Discrete Mathematics

Download Propositions - Discrete Mathematics and its Applications - Lecture Slides and more Slides Discrete Mathematics in PDF only on Docsity!

Discrete Mathematics

Math 6A

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”.

P

¬

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.

F F T T

F T T F

T F F F

T T T T

P Q P → Q P ↔ Q

weird?

Consider: P  Q. converse: Q  P. contra-positive: (NOT Q)  (NOT P) (equiv.) inverse: (NOT P)  (NOT Q).

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!

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.

n

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

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:

∃ x Q ( x , y )

∃ x ( P ( x )∧ Q ( x )) ∨ ∀ y R ( y )

scope x scope y

( ) ( )

( ) ( )

xP x x P x

xP x x P x

¬∃ ≡∀ ¬

¬∀ ≡∃ ¬

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 xDocsity.com