Untitled document (1).pdf

COVERAGE FOR PRELIMS A statement is a declarative sentence that is either true or false, but not both true and false. A simple statement is a statement that conveys a single idea. A compound statement is a statement that conveys two or more ideas. - Connecting simple statements with words and p...

COVERAGE FOR PRELIMS A statement is a declarative sentence that is either true or false, but not both true and false. A simple statement is a statement that conveys a single idea. A compound statement is a statement that conveys two or more ideas. - Connecting simple statements with words and phrases such as and, or, if... then, and if and only if creates a compound statement. Logic Connectives The truth value of a simple statement is true if is a true statement, and the truth value of a simple statement is false if it is a false statement. - The truth value of a compound statement depends on the truth values of its simple statements and its connectives. - NOTE! If a statement in symbolic form is written as an English sentence, then the simple statements that appear together in parentheses in the symbolic form will all be on the same side of the comma that appears in the English sentence. The truth table below shows the four possible cases that arise when we form a conjunction of two statements Quantifiers and Negation In a statement, the word some and the phrases there existsand at least one are called existential quantifiers. Existential quantifiers are used as prefixes to assert the existence of something. In a statement, the words none, no, all, and every are called universal quantifiers. The universal quantifiers none and no deny the existence of something, whereas the universal quantifiers all and every are used to assert that every element of a given set satisfies some condition. Truth Tables, Equivalent Statements, and Tautologies – The Conditional and the Biconditional Conditional In any conditional statement represented by "If p, then q" or by "If p, q," the p statement is called the antecedent and the q statement is called the consequent. The conditional statement, "If p, then q," can be written using the i." a * " p —> q. The arrow notation p —> q is read as "if p, then q" or as "p implies q." Because p > q =~ pVq, an equivalent form of ~(p -> q) is given by~(~pVq), which by one of De Morgan's laws, can be expressed as the conjunction pA ~ q. Hence, ~ (p> q) = pA ~ q. Biconditional Definition 3.3.1 The statement (p->q) /(q->p) is called a biconditional and is denoted by p q which is read as "p if and only if q." Hence, p q= [(p>q) A(q->p)] Equivalent Forms of the Conditional Every conditional statement can be stated in many equivalent forms. It is not even necessary to state the antecedent before the consequent. For instance, the conditional "If I live in Luzon, then I must live in Manila" can also be stated as I must live in Manila, if I live in Luzon. The Converse, the Inverse, and the Contrapositive Every conditional statement has three related statements. They are called the converse, the inverse, and the contrapositive. Statements Related to the Conditional Statement The converse of p -> q is q -> p. The inverse of p -> q is ~p -> ~q. The contrapositive of p -> q is ~q -> ~p The previous definitions show the following: The converse of p -> q is formed by interchanging the antecedent p with the consequent q. The inverse of p p -> q is formed by negating the antecedent p and negating the consequent q. The contrapositive of p -> q is formed by negating both the antecedent p and the consequent q and interchanging these negated statements.