3.2. Basic Laws of Logical Operations

1. Idempotency of disjunction and conjunction:

F V F ^ F, F A F ^ F.

2. Commutativity of disjunction and conjunction:

F V F ^ F V F, F A F ^ F A F.

3. Associativity of disjunction and conjunction:

F AF ^FVF,

F v(F V ZF V F)v Z, F A(F A ZF A F)A Z .

4. Double negation F ^ F .

5. De Morgan laws:

V

F V F ^ F A F,

6. Distributivity of disjunction and conjunction operations with respect to each other: F V(F AZ)^(F VF)A(F VZ); F A(F V Z)^(F A F)V(F A Z).

7. Sewing:

(F V F)A(F V F F;

7. Absorption:

(F A F)V(F A F F

F A(F V FF .

F V(F AFF;

9. Operations with logical constants 0 and 1:

F A F ^ 0;

F V 0 ^ F; F A 0 ^ 0;

F V F ^ 1.

F ^ F.

F A F ^ 1.

F V1 ^ 1; F A1 ^ F.

10. Law of the excluded middle:

11. Identity:

12. Negation of contradiction:

13. Contraposition: (F ^ F) ^ (F ^ F).

14. Chain rule: (F ^ F)A(F ^ Z)^(F ^ Z).

15. Antithesis: (F ^ F) ^ (F ^ F).

16. Modus ponens, which means "proposing mode": F A(F ^ FF .

^..^w^/e 3.2.7. Suppose that the proposition F is "it is raining" and the proposition F is "cats and dogs get wet", then the compound proposition " it is raining; and if it is raining, then cats and dogs get wet" logically implies that cats and dogs are really wet.

13. Modus tollense, which means "removing mode":

(F ^ F)A(FF . As can be seen, it is a counterpart of modus ponence. For instance, in the previous example we just used for modus ponence, modus tollense would state: the compound proposition " if it is raining; then cats and dogs get wet, and cats and dogs are not wet " which logically implies that it is not raining.

3.3. Propositiona! Functions. Quantifiers

Let ^ be given set. A propositional function (or an open sentence or condition) defined on ^ is an expression ^(.x), which has the property that ) is true or false for each й E ^. That is, ^(.x) becomes a statement (with a truth value) whenever any element й E ^ is substituted for the variable .x. The set ^ is called the domain of ^(.x), and the set of all elements of ^ which ^(й) is true is called the

truth set of ^(.x). In other words,

= {.[ .x E ^(.x) is true} or = {.] ^(.x) }. Frequently, when ^ is some set of numbers, the condition ^(x) has the form of an equation or inequality involving the variable . .

Е.й^^/е 3.3.7. Find the truth set for each propositional function ^(x) defined on the set jV :

1. Let ^(x) be "x + 2 > 7". Its truth set is {б, 7,8,...} consisting of all integers greater than 5.

2. Let ^(x) be " x + 5 < 3 ". Its truth set is the empty set 0. That is, ^(x) is not true for any integer in V.

3. Let ^(x) be " x + 5 > 1". Its truth set is V. That is, ^(x) is true for every element in V.

Let ^(x) be a propositional function defined on a set Consider the expression

(Vx E ^(x) or Vx^(x) (3.3.1)

which reads "For every x in ^(x) is a true statement" or, simply, "For all x, ^(x)". The symbol V which reads "for all" or "for every" is called the universal quantifier. The statement (3.3.1) is equivalent to the statement

Г^ ={x]x E ^(x )}= ^ (3.3.2)

that is, that the truth set of ^(x) is the entire set

The expression ^(x) by itself is an open sentence or condition and therefore has no truth value. However, V.xp(x), that is ^(x) preceded by the quantifier V, does have a truth value which follows from the equivalence of (3.3.1) and (3.3.2). Specifically: If {x x E ^(x)}= ^ then V.xp(x) is true, otherwise, V.xp(x) is false.

Е.й^^/е 3.3.2.

1. The proposition (V^ E V) (и + 4 > 3) is true since {и и + 4 > 3} = {1,2,3,...} = V.

2. The proposition (VM E V)(и + 2 > 8) is false since {и] и + 2 > 8}= {7,8,9,...}^ V.

3. The symbol V can be used to define the intersection of an indexed collection

E /} of sets as follows:

E /)={x[ V'' E /, x E^. }.

Let ^(x) be a propositional function on a set Consider the expression

(3x E ^(x) or 3 x ^(x) (3.3.3)

which reads "There exists an x in ^ such that ^(x) is a true statement" or, simply, "For some x, ^(x)". The symbol 3 which reads "there exists" or "for some" or "for at least one" is called the existential quantifier. The statement (3.3.3) is equivalent to the statement

={x]x E ^(x )}# 0 (3.3.4)

i.e., that the truth set of ^(x) is not empty. Accordingly, 3x ^(x), that is ^(x) preceded by the quantifier 3, does have a truth value. Specifically: If {x x E ^(x)}# 0 then 3xp(x) is true, otherwise, 3xp(x) is false.

Exaw^/e 3.3.2.

1. The proposition (3M E A^)(M + 4 < 7) is true since {M M + 4 > 3}= {1,2}^0.

2. The proposition (3 M EA) (M + 6 < 4) is false since {M M + 6 < 4}= 0.

3. The symbol 3 can be used to define the union of an indexed collection ' E /} of sets as follows:

u(4.['' E /)={x 3'' E /, x E^. }.