implication of mathematical prepositions

1. Presented by:
Varsha
Mathematical prepositions:
implications- necessary and sufficient
conditions.

2. Implication:
 A proposition is a statement that is either true or
false.
 Mathematical propositions are often of the form “If
P, then Q”. Such a statement is called an
implication. “P” is called the hypothesis and “Q” is
called the conclusion.
 For example, “If x and y are odd integers, then xy
is an odd integer.” If the hypothesis is false, then
the implication is called true no matter whether
the conclusion is true or false.

3.  The following is a true proposition: “If cows could
fly then the moon is made of blue cheese.” The
implication “if P then Q” is sometime written as or
“Q whenever P”.
P Q P→Q
T T T
T F F
F T T
F F T

4.  p → q is read in a variety of equivalent ways:
 if p then q
 p only if q
 p is sufficient for q
 q whenever p

5.  Examples: – if sun rise from west then 2 is a
prime.
 If F then T ?
 True
 if today is Tuesday then 2 * 3 = 8. If T then F?
 False

6. Converse of implications:
 The converse of a proposition “if P then Q” is the
statement “if Q then P”. The converse of a
statement is not equivalent to a statement. For
example the proposition “if x is positive then x 2 is
positive” is true, while “if x 2 is positive then x is
positive” is false.
 The converse of p q is q p

7.  Examples:
 If it snows, the traffic moves slowly.
 p: it snows q: traffic moves slowly.
 p q – The converse: If the traffic moves slowly
then it snows.
 q p

8. Contra- positive of Implication
 The contra-positive of an implication is
an implication with the antecedent
and consequent negated and interchanged. For
example, the contra-positive of (p ⇒ q) is (¬q ⇒ ¬p).
Note that an implication and it contra-positive are
logically equivalent.

a proposition or theorem formed by contradicting both
the subject and predicate or both the hypothesis and
conclusion of a given proposition or theorem and
interchanging them "if not-B then not-A " is
the contra-positive of "if A then B “
 • If it snows, the traffic moves slowly. – The contra-
positive: • If the traffic does not move slowly then it
does not snow. • ¬q ¬p – The inverse:

9. Inverse of Implication:
 The inverse of the conditional statement is “If
not P then not Q.”
 The inverse of p q is ¬p ¬q
 Example:– If it snows, the traffic moves slowly.
 The inverse: • If it does not snow the traffic moves
quickly. • ¬p ¬q