Statements

Solution:

A statement is any declarative sentence for which it makes sense to consider whether it is true or false. We denote statements by: A, B, C, D, P, ... etc.

Statement

• a)    true – “holds,” has the truth value (1)
• b)    false – “does not hold,” has the truth value (0)

Negation of a statement

For every statement A, we can form a statement A‘ that denies (negates) what statement A asserts. Statement A‘ is called the negation of statement A. We form the negation by putting before the statement a prefix like “not ...,” “is not,” or the text “it is not true that ....”

Solution:

A': The number 3 is not a prime number (0)

B': Bratislava does not lie in Egypt (1)

C : not a statement

D': Mathematics is not a science (0)

E': The number 22 is not divisible by 2 (0)

F : not a statement

G : cannot be decided

H': Sine of 300 is not −2.1 (1)

I : not a statement

J : not a statement

K': The number 2574364 is not divisible by 4 (0)

L': It is not true that (a + b)² = a² + b² (1)

M': The area of a circle is not S = π·r² (0)

N': It is not true that 4² equals 18 (1)

O': I do not have a new car (0)

P : not a statement

Solution:

A quantified statement is a declarative sentence that gives a certain number, or an estimate of the number, of objects, persons, etc., with the same property. In a quantified statement we encounter words such as: exactly, at most, every, all, some, at least, none ... etc., which are called quantifiers, as well as numerals.

• The statement “at least 5” means 5 or more.
• The statement “at most 5” means 5 or less.

For symbolic notation of quantified statements we use

• a)    the universal quantifier  – “for every (all) it holds that.....”
• b)    the existential quantifier  – “there exists at least one ... for which it holds that ....”

Negation of a quantified statement