Logic and propositional calculus

Logic and propositional calculus
A proposition (or statement) is a declarative statement which is true or false, but not both. Example: the following six sentences:
(1) Ice floats in water.
(2) China is in Europe. (3) 2 + 2 = 4
(4) 2 + 2 = 5
(5) Where are you going?
(6) Do your homework.
The first four are propositions, the last two are not. Also, (1) and (3) are true, but (2) and (4) are false.
Compound Propositions
It is the proposition that composed of sub propositions and various connectives.
Primitive proposition is the proposition that cannot be broken down into simpler propositions. For example, the above propositions are primitive propositions, while:
“Roses are red and violets are blue.” and
“John is smart or he studies every night.”, Are compound.
BASIC LOGICAL OPERATIONS
1- Conjunction, p ^ q 2- Disjunction, p v q 3- Negation, ?p
EXAMPLE : Consider the following four statements:
(i) Ice floats in water and 2 + 2 = 4.
(ii) Ice floats in water and 2 + 2 = 5.
(iii) China is in Europe and 2 + 2 = 4.
(iv) China is in Europe and 2 + 2 = 5.
Only the first statement is true. Each of the others is false since at least one of its substatements is false.

EXAMPLE: Consider the following four statements:
(i) Ice floats in water or 2 + 2 = 4.
(ii) Ice floats in water or 2 + 2 = 5.
(iii) China is in Europe or 2 + 2 = 4.
(iv) China is in Europe or 2 + 2 = 5.

Only the last statement (iv) is false. Each of the others is true since at least one of its sub- statements is true.

EXAMPLE : Consider the following six statements:
(a1) Ice floats in water. (a2) It is false that ice floats in water. (a3) Ice does not float in water. (b1) 2 + 2 =5 (b2) It is false that 2 + 2 = 5. (b3) 2 + 2 ?5

Then (a2) and (a3) are each the negation of (a1); and (b2) and (b3) are each the negation of (b1). Since (a1) is true, (a2) and (a3) are false; and since (b1) is false, (b2) and (b3) are true.

PROPOSITIONS AND TRUTH TABLES

The truth table for the compound proposition ?(p??q) is:


TAUTOLOGIES AND CONTRADICTIONS
Some propositions P(p, q, . . .) contain only T in the last column of their truth tables or, in other words, they are true for any truth values of their variables. Such propositions are called tautologies. Analogously, a proposition P(p, q, . . .) is called a contradiction if it contains only F in the last column of its truth table or, in other words, if it is false for any truth values of its variables. For example, the proposition “p or not p,” that is, p ??p, is a tautology, and the proposition “p and not p,” that is, p??p, is a contradiction.


Note that the negation of a tautology is a contradiction since it is always false, and the negation of a contradiction is a tautology since it is always true.

LOGICAL EQUIVALENCE
Two propositions P(p, q, . . .) and Q(p, q, . . .) are said to be logically equivalent, or equal, denoted by P(p, q, . . .) ? Q(p, q, . . .) if they have identical truth tables.


for example, the truth tables of ?(p ? q) and ?p??q, both truth tables are the same, that is, both propositions are false in the first case and true in the other three cases. Accordingly, we can
write: ?(p ? q) ? ?p ??q

ALGEBRA OF PROPOSITIONS


CONDITIONAL AND BICONDITIONAL STATEMENTS
Many statements, particularly in mathematics, are of the form “If p then q.” Such statements are called conditional statements and are denoted by: p ? q
The conditional p ? q is frequently read “p implies q” or “p only if q.”
Another common statement is of the form “p if and only if q.” Such statements are called biconditional statements and are denoted by p ? q


(a) The conditional p ? q is false only when the first part p is true and the second part q is false. Accordingly, when p is false, the conditional p ? q is true regardless of the truth value of q.
(b) The biconditional p ? q is true whenever p and q have the same truth values and false otherwise.
Note that the truth table of p ? q and ?p ?q and are identical, that is, they are both false only
in the second case. Accordingly, p ? q is logically equivalent to ?p ? q; that is, p ? q ? ?p ? q