• Conditional Statements
• The Negation of a Conditional Statement
• Contrapositive
• Converse and Inverse
• Only if and the Biconditional
• Necessary and Sufficient Conditions

Conditional Statements: Let p and q be statements. A sentence of the form "If p, then q" is denoted symbolically by "p ® q"; p is called the hypothesis, premise, or antecedent and q is called the conclusion or consequence. The symbol ® (if-then) is a binary connective, like Ù and Ú , that can be used to join statements to create new statements. Referred to as if-then or as a conditional. Most computer programs with any degree of difficulty utilize this connector to make alternative decisions. The truth table for ® (if-then) is intuitive.

Suppose you go to interview for a job and are promised:

If you show up for work Monday morning, then you will get the job.

Under what circumstances are you justified in saying the promise was broken and the statement false? Answer: you do show up for work Monday morning and you do not get the job. Certainly, if you do not show up and do or do not get the job you can not say the promise was broken.

Definition:
If p and q are statement variables, the conditional of q by p is "If p, then q" or "p implies q" and is denoted p® q. It is false when p is true and q is false; otherwise it is true. The conditional, ®, has lower precedence than the conjunction and disjunction operators.

The above truth table indicates that the truth value of the conditional will be the same as when Ø p is true, or when q is true. This combination is the same as Ø p Ú q.

The foregoing analysis with the conditional allows us a means of determining a statement for any given truth table. This method is known as disjunctive normal form.

Example:
Determine a statement using disjunctive normal form satisfying the below truth table:

Note from line 3 that Ø p Ù q produces a value of T. Line 4, Ø p Ù Øq, also produces a true value. Thus we get a value of T either from line 3 or from line 4 (or both). The required statement is formed from the disjunction of the two rows.

(Ø p Ù q) Ú (Ø p Ù Øq).

 Example:
Find a statement, using disjunctive normal form, having the truth values as indicated below:

(p Ù Ø q) Ú (Ø p Ù q).

p ® q Û Ø p Ú q.

Use truth tables to establish the logical equivalence of "if p, then q" and "not p or q."

Negation of a Conditional Statement top

By definition p ® q is false if, and only if, its hypothesis, p, is true and its conclusion, q, is false. It follows that the negation of "if p, then q" is logically equivalent to "p and not q."

Ø ( p ® q) Û p Ù Ø q.

This result can be obtained using logical equivalence

 The Contrapositive of a Conditional Statement top
Definition:

The contrapositive of a conditional statement of the form "If p, then q" is

If Ø q, then Ø p.

Symbolically,

The contrapositive of p® q is Ø q® Ø p.

A conditional statement is logically equivalent to its contrapositive.

The contrapositive form is frequently used in mathematics and computer science to make solutions simpler or to offer another approach to a tough problem. This logical equivalence is the basis for one of the most important laws of deduction, modus tollens, and for the contrapositive method of proof.

Example:

Write "If the Cubs win the pennant, then they will have won tomorrows game." in contrapositive form.

If the Cubs do not win tomorrows game, then they will not win the pennant.

The Converse and Inverse of a Conditional Statement top
Definition:
Suppose a conditional statement of the form "If p, then q" is given,

1. The converse is "If q, then p."
2. The inverse is "If Ø p, then Ø q."

Symbolically,

the converse of p® q is q® p,

and

The inverse of p® q is Ø p® Ø q.

Careful:

1. A conditional statement and its converse are not logically equivalent.
2. A conditional statement and its inverse are not logically equivalent.
3. The converse and the inverse of a conditional statement are logically equivalent to each other.

Definition: top
If p and q are statements,

p only if q means "if not q, then not p,"

or, equivalently (by its contrapositive)

"if p, then q."

There are many synonyms for the if-then statement in English and a number of these are used equivalently in mathematical reasoning. p ® q can be expressed as:

1. If p, then q.
2. q if p.
3. q by p.
4. p only if q.
5. q follows from p.
6. q provided p.
7. q whenever p.
8. q is a necessary condition for p (q follows necessarily from p).
9. p is a sufficient condition for q.

If it is known that both p ® q and its converse, q ® p, are true, we know that q follows from p and p follows from q. This gives rise to our next binary connective, the biconditional, denoted p if, and only if q, (p Û q). Some authors use the term equivalence for this connector. It is true if both p and q have the same truth values and is false if p and q have opposite truth values.

Necessary and Sufficient Conditions top
Definition:

If r and s are statements:
r is a sufficient condition for s means "if r, then s."
r is a necessary condition for s means "if not r, then not s."

Example:

Rewrite the following statement as an implication in two ways (utilize the contrapositive):
George's attaining age 35 is a necessary condition for his being president of the United States.
Answer.