logic and proposition

7/30/2019 Logic and Proposition

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Logic and PropositionalCalculus

Engr. Francis A. MalabananECE, Instructor


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Aproposition(or statement) is a declarativestatement which is true orfalse, but not both.

i. Ice floats in water.ii. China is in Europe.

iii. 2 + 2 = 4

iv. 2 + 2 = 5v. Where are you going?

vi. Do your homework.


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Many propositions are composite, that is,composed ofsubpropositions and various

connectives discussed subsequently. Suchcomposite propositions are called compoundpropositions.

A proposition is said to beprimitive if itcannot be broken down into simplerpropositions, that is, if it is not composite.


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Primitive proposition: Ice floats in water.

China is in Europe. 2 + 2 = 4

2 + 2 = 5

Compound proposition: Roses are red and violets are blue.

John is smart or he studies every night.


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3 Basic logical operations

Negation,p

Disjunction, p q

Conjunction, p q


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Letp be a proposition. The negation of p,

denoted byp (also denoted by p), is the

statement It is not the case that p. The propositionp is read not p. The truth value of the negation of p,p,

is the opposite of the truth value of p.If p is true, then p is false; and if p is false,

then p is true.


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a1) Ice floats in water.a2) It is false that ice floats in water.

a3) Ice does not float in water.b1) 2 + 2 =5b1) It is false that 2 + 2 = 5.

b1) 2 + 2 5


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Any two propositions can be combined by theword or to form a compound proposition

called the disjunctionof the originalpropositions. Symbolically,p q

The proposition p q is read p or q,denotesthe disjunction of p and q.

If p and q are false, then p q is false; otherwise p q is true.


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Ice floats in water or 2 + 2 = 4.China is in Europe or 2 + 2 = 4.

Ice floats in water or 2 + 2 = 5.

China is in Europe or 2 + 2 = 5.


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Any two propositions can be combined by theword and to form a compound proposition

called the conjunction of the originalpropositions. Symbolically,p q

The proposition p q is read p and q,denotes the disjunction of p and q.

If p and q are true, then p q is true; otherwise p qis false.


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Ice floats in water and 2 + 2 = 4.China is in Europe and 2 + 2 = 4.

Ice floats in water and 2 + 2 = 5.

China is in Europe and 2 + 2 = 5.


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Let P(p, q, . . .) denote an expression

constructed from logical variablesp, q, .

. ., which take on the value TRUE (T) orFALSE (F), and the logical connectives ,, and . Such an expression P(p, q, . . .)

will be called aproposition.


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The actual truth table of the proposition

(p q)


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Remarks: In order to avoid an excessivenumber of parentheses, we sometimes adopt

an order of precedence for the logicalconnectives. Specifically,

has precedence over which hasprecedence over

For example:

p q means (p) q and not (p q)


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j p q p j q1 2

3 4 5

6

Simply: ( j p ) ( q ( p )) ( j ( q ))


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indicates how the truth table of (p q) is

constructed


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Note: For 2 variables, as above, 4 rowsare necessary; for 3 variables, 8 rows

are necessary; and, in general, for nvariables, 2n rows are required.


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For three proposition j, p, and qwe have 23 combinations (8)

j p q ( j p q )T T T F

T T F F

T F T F

T F F TF T T F

F T F T

F F T F

F F F T

12

3

456

7

8


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Use the laws to show that

(p q) (p q) p

Statement Reason1. (p q) (p q) (p q) (p q)2. p (q q)3. p T4. p


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Letp and q be propositions. The conditionalstatementp q is the proposition if p, then

q. The conditional statementp

q is falsewhenp is true and q is false, and trueotherwise.


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Letp and q be propositions. The biconditional

statementp q is the proposition p if and

only if q. The biconditional statementp

qis true whenp and q have the same truthvalues, and is false otherwise.


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Prepared by:

Engr. Francis A. Malabanan

ECE, Instructor

First Asia Institute of Technology and Humanities

logic and proposition

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