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 ® (ifthen) is a binary connective, like Ù and Ú , that can be used to join statements to create new statements. Referred to as ifthen or as a conditional. Most computer programs with any degree of difficulty utilize this connector to make alternative decisions. The truth table for ® (ifthen) 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"^{1} 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.
p 
q 
p ® q 
T 
T 
T 
T 
F 
F 
F 
T 
T 
F 
F 
T 
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:
p 
q 
? 
1. T 
T 
F 
2. T 
F 
F 
3. F 
T 
T 
4. F 
F 
T 
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 
q 
? 
1. T 
T 
F 
2. T 
F 
T 
3. F 
T 
T 
4. F 
F 
F 
(p Ù Ø q) Ú (Ø p Ù q).
Representation of Ifthen as Or
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
Ø ( p ® q) 
Û 
Ø (Ø p Ú q) 


Û 
Ø (Ø p) Ù Ø q 
DeMorgan's law 

Û 
p Ù Ø q 
double negative law 
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 1
Write "If the Seahawks win the CAA tournament, then they will have won tomorrows game." in contrapositive form.
Soln: If the Seahawks do not win tomorrows game, then they will not win the CAA tournament.
The Converse and Inverse of a Conditional Statement top
Definition:
Suppose a conditional statement of the form "If p, then q"
is given,
Symbolically,
the converse of p® q is q® p,
and
The inverse of p® q is Ø p® Ø q.
Careful:
Example 2
Conditional 
Converse 

If it is cold, then I will wear my coat
If you fall in the pond, then you will get wet. I'll kiss you if the moon is made of green cheese. 
If I wear my coat, then it is cold.
If you are wet, then you fell in the pond. If I kiss you, then the moon is made of green cheese. 

It should be obvious in the examples above that the converse does not have the same meaning as the conditional. 
Example 3
Example 4 Try It
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 ifthen statement in English and a number of these are used equivalently in mathematical reasoning. p ® q can be expressed as:
Biconditional (Û )
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."
Consider the statement "If John is eligible to vote, then he is at
least 18 years old." Let r be
"John is eligible to vote" and let s
be "John is at least 18 years old." The truth of the condition "John is
eligible to vote" is sufficient to
ensure the truth of the condition "John is at least 18 years old." That
is, if r then s.
In addition, the condition "John is at least 18 years old" is necessary for
the condition "John is eligible to vote" to be true. If John were
younger than 18, then he would not be eligible to vote. Which can be
seen as if not r, then not s.
http://faculty.uncfsu.edu/jyoung/necessary_and_sufficient_conditions.htm
1. The philosopher Willard VanOrman Quine advises against using the phrase "p implies q" to mean "p → q" because the word implies suggests that q can be logically deduced from p and this is often not the case. The phrase is used by many people, probably because it is convenient.
2. Discrete Structures: an introduction to mathematics for computer science, Fletcher R. Norris, PrenticeHall, pp 2829.