Statements and Quantifiers - Contemporary Math Chapter 2.1

Logical statements Negating statements Expressing statements with quantifiers Negating quantifiers
Statements and quantifiers
Contemporary Matehmatics
Chapter 2.1
Dr Wilmari Morton
Dr Wilmari Morton
Statements and quantifiers

Logical statements Negating statements Expressing statements with quantifiers Negating quantifiers
Identifying logical statements
Alogical statement is a complete declarative sentence that makes
a claim that can be determined to be either true or false (but not
both).
Example
1The sky is blue.
2Are you John?
3Please, keep quiet.
4This circle is a square.
56 is a real number.
6The square root of 4 is 2.
7This sentence is false.
Dr Wilmari Morton
Statements and quantifiers

Logical statements Negating statements Expressing statements with quantifiers Negating quantifiers
Representing logical statements symbolically
Symbolic form is used to reduce the amount of writing involved.
Symbolic form also helps visualize the relationship between the
statements in a more concise way in order to determine the
strength or validity of an argument. Each logical statement is
represented symbolically as a single lowercase letter, for example
p,qor ror p1,p2. . ..
Example
1Let prepresent “The sky is blue.”
2Let qrepresent “3 plus 2equals 6.”
3Let p1represent “4 times 3equals 12.”
Dr Wilmari Morton
Statements and quantifiers

Logical statements Negating statements Expressing statements with quantifiers Negating quantifiers
The negations of a logical statement
The negation of a logical statement has the opposite truth value of
the original statement.
Example
1The negations of “The sky is blue” is “The sky is not blue”
2The negation of “3 plus 2equals 6” is
“3 plus 2 does not equal 6”.
3The negations of “4 times 3equals 12” is
“4 times 3 does not equal 12”.
4The negation of “The circle is not a square” is
“The circle is a square”.
Dr Wilmari Morton
Statements and quantifiers

Logical statements Negating statements Expressing statements with quantifiers Negating quantifiers
Negating symbolically
The symbol for negation, or not, in logic is the tilde, ∼.
Example
If prepresents “πis a natural number”, then ∼prepresents “πis
not a natural number”.
Dr Wilmari Morton
Statements and quantifiers