statements and connectives
A statement is a sentence or phrase that must have a precise mathematical meaning.
A simple statement has a truth value of TRUE or FALSE but not both.
Here are some simple statements:
- Any square is also a rectangle
- Any rectangle is also a square
- 1 + 1 = 3
- Syria is in the Middle East
A COMPOUND STATEMENT is made up of simple statements joined together by CONNECTIVES.
The five connectives we most commonly use are:
NOT, AND, OR, OR*, IF..THEN
*There are two versions of OR - Exclusive OR - p or q but not both
- Inclusive OR - p or q or both
A simple statement has a truth value of TRUE or FALSE but not both.
Here are some simple statements:
- Any square is also a rectangle
- Any rectangle is also a square
- 1 + 1 = 3
- Syria is in the Middle East
A COMPOUND STATEMENT is made up of simple statements joined together by CONNECTIVES.
The five connectives we most commonly use are:
NOT, AND, OR, OR*, IF..THEN
*There are two versions of OR - Exclusive OR - p or q but not both
- Inclusive OR - p or q or both
truth tables
Practice:
Related past paper questions
implications - Converse, inverse and contrapositive
An “If ……. then…….” statement is called an implication and we write p ⟹ q.
Converse -
The converse of the statement p ⟹ q is the statement q ⟹ p.
Inverse -
The inverse statement p ⟹ q is the statement ¬p ⟹ ¬q.
Contrapositive -
The contrapositive statement p ⟹ q is the statement ¬q ⟹ ¬p.
Converse -
The converse of the statement p ⟹ q is the statement q ⟹ p.
Inverse -
The inverse statement p ⟹ q is the statement ¬p ⟹ ¬q.
Contrapositive -
The contrapositive statement p ⟹ q is the statement ¬q ⟹ ¬p.
Equivalence -
Two statements are equivalent when p ⟹ q AND q ⟹ p.
This can be written as p ⟺ q and can be spoken as “ if and only if” (iff).
Use this information to construct a truth table for p ⟺ q.
Two statements are equivalent when p ⟹ q AND q ⟹ p.
This can be written as p ⟺ q and can be spoken as “ if and only if” (iff).
Use this information to construct a truth table for p ⟺ q.
Logical equivalence
Equivalence depends only on the structure of the two compound statements. It does not depend on the meaning of the initial statements p and q.
Examples:
Examples:
The entries in the last columns in the two truth tables in both examples are exactly the same. These identical entries tell us that whatever the truth values of p and q, the compound statements ¬p ⋀ ¬q and ¬(p v q) have the same truth values. There is no logical difference between them.
tautologies and contradictions
If we have a tautology, we can say that our argument is valid. This means that the conclusion we have made follows logically from the premise.
valid and Invalid arguments:
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