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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.

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Discrete mathematics math 6a

Discrete MathematicsMath 6A

Instructor: M. Welling


1 1 propositions
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”.


1 1 propositions1
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).


1 1 implications
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?


1 1 precedence bits
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
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:


1 2 propositional equivalences1
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.


1 3 predicates quantification
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 predicates and quantification
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)


1 3 binding negations
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


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