Logical Reasoning
Mathematics SSS 2 Second Term
WEEK 2
Logical Reasoning
Performance Objectives
Students should be able to;
- Use the truth table to prove that a contrapositive is equivalent to a conditional statement.
- Use truth table to prove that a converse is equivalent to an inverse of a conditional statement
- Apply contrapositive and inverse in proving theories
Conditional statements
Content
In mathematics, many statements are of the form 'If P, then Q’. Such statements are called conditional statements or implications.
When two simple statements are combined by 'If ...... then', such that the first statement implies the second. They are denoted by P =» Q. The if clause statement i.e. P) is sometimes called the antecedent while the then clause (Q) is called the consequent.
The conditional statement P =» Q can also be read as:
a) P implies Q
b) Q only if P
c)P is sufficient for Q
d) P is necessary for Q
Example
- if P stands for 'Pastors preach' and Q stands for ‘Pastors talk a lot’.
Then P =» Q is the statement ‘If Pastor Preach, then he talks a lot’
The truth value of P =» Q for Conditional statements
|
P |
Q |
P =» Q |
|
T |
T |
T |
|
T |
F |
F |
|
F |
T |
T |
|
F |
F |
T |
Properties of P =» Q
- Converse of P =» Q is Q =» P
- Inverse of P =» Q is ˜P =» ˜Q
- Contra-positive of P =» Q is ˜Q =» ˜P
Note: the P =» Q column is false only when P is true and Q is false.
Bi-Conditional Statement
If P statement is =» Q statement and Q statement is =» P, such a relation implies equivalence.
Example
If P stands for ‘john speaks English fluently’ and Q stands for ‘john was born and bred in England’, then P ó Q.
Truth table for Bi-Conditional Statement
|
P |
Q |
P =» Q |
|
T |
T |
T |
|
T |
F |
F |
|
F |
T |
T |
|
F |
F |
T |
Example
Determine the truth values of the following compound statements.
- ˜(P ˄ ˜Q)
- ˜P => Q
- (P ˅ Q) ó (P ˅ ˜Q)
Solution
- ˜(P ˄ ˜Q)
|
P |
Q |
˜Q |
P ˄ ˜Q |
˜(P ˄ ˜Q) |
|
T |
T |
F |
F |
T |
|
T |
F |
T |
T |
F |
|
F |
T |
F |
F |
T |
|
F |
F |
T |
F |
T |
- ˜P => Q
|
P |
˜P |
Q |
˜P =>> Q |
|
T |
F |
T |
T |
|
T |
F |
F |
T |
|
F |
T |
T |
T |
|
F |
T |
F |
F |
C. (P ˅ Q) ó (P ˅ ˜Q)
|
P |
Q |
˜Q |
P˅Q |
P˅˜Q |
(P˅Q) ó (P˅˜Q) |
|
T |
T |
F |
T |
T |
T |
|
T |
F |
T |
T |
T |
T |
|
F |
T |
F |
T |
F |
F |
|
F |
F |
T |
F |
T |
F |
Contradiction
This is a situation where a compound statement results to false despite the validity or otherwise of the substatements. Contradiction is always denoted by F
Example
Show that the statement (P ˄ Q) ˄ ˜(P ˅ Q) is a contradiction
Solution
|
P |
Q |
P ˄ Q |
P ˅ Q |
˜(P ˅ Q) |
(P˄Q)˄ ˜(P˅Q) |
|
T |
T |
T |
T |
F |
F |
|
T |
F |
F |
T |
F |
F |
|
F |
T |
F |
T |
F |
F |
|
F |
F |
T |
F |
T |
F |
From the table above it can be seen that the truth values for the last column of the truth table is all false, it means all the statement is a contradiction.
Note the following
- A compound statement is a statement which consists of two or more simple statements or substatements.
- The four logical connections are ‘and’, ‘or’, if…….then’ and ‘if and only if’
- The symbols for negation and the other connectives are
|
S/N |
CONNECTIVE |
WORD |
SYMBOLS |
|
|
Conjunction A |
‘And’ |
˄ |
|
|
Disjunction |
'or' |
V |
|
|
Negation |
'not’ |
˜ |
|
|
Implication |
‘Implies’ |
=> |
|
|
Bi-implication |
‘if and only if’ |
ó |
|
|
Equivalence |
‘Is’ |
Ξ |
- The truth table technique is one of the ways of establishing the validity of any statement in logic
- Two compound statements are said to be logically equivalent if they have the same truth value
- A composite statement which is true for all truth values of the substatement it involves is called a Tautology
- A compound statement which is false for all truth values of the statements it involves is called a Contradiction (F)