The document outlines key concepts in mathematical logic focusing on negation and quantifiers. It explains how negation reverses the truth value of statements and includes important rules for manipulating logical expressions. Additionally, it discusses the importance of consistency in axiomatic systems and clarifies the nature of absurd statements and contradictions.

1. 8
8
BSED MATH
BSED MATH

2. LOGIC AND INCIDENCE
LOGIC AND INCIDENCE
GEOMETRY
GEOMETRY
NEGATION AND
NEGATION AND
QUANTIFIERS
QUANTIFIERS

3. LEARNING OBJECTIVES
LEARNING OBJECTIVES
At the end of the lesson , learners will be able to:
K- Define negation as a logical operation that reverses the
truth value of a statement.
S- Apply the rules og negation to manipulate and transform
mathematical statements and expressions
A- Appreciate the significance of negation and quantifiers
in formal mathematical reasoning and proof-writing.

4. First, some remarks on notation. If S is any
statement, we will
denote the negation or contrary of S by ~S.
For example, if S is the
statement "p is even," then ~S is the statement
"p is not even" or "p is odd.”
NEGATION

5. LOGIC RULE
The statement "~(~S)" means the same as "S."
We followed this rule when we negated the statement
“ 2 is irrational” by writing the contrary as “ 2
is rational” instead of “ 2 is not irrational.”
NEGATION
NEGATION

6. QUANTIFIERS
QUANTIFIERS
We wish to prove H => C, and we assume, on the
contrary, H does not imply C, i.e., that H holds
and at the same time ~C holds. We write this
symbolically as H & ~C, where
& is the abbreviation for "and." A statement
involving the connective
"and" is called a conjunction. Thus:

7. QUANTIFIERS
QUANTIFIERS
LOGIC RULE 4.
The statement "~[H => C] " means the same as "H &
~C.”
LOGIC RULE 5.
The statement "-[S1 & S2]" means the same as
“[~S1 or ~S2].”

8. Finally let us be more precise about what is an absurd statement. It
is the conjunction of a statement S with the negation of S, i.e., "S &
~S." A statement of this type is called a contradiction. A system of
axioms from which no contradiction can be deduced is called consistent.
QUANTIFIERS
QUANTIFIERS

9. QUANTIFIERS
QUANTIFIERS

10. QUANTIFIERS
QUANTIFIERS
It must be emphasized that a statement beginning with "For
every . . ." does not imply the existence of anything.

11. QUANTIFIERS
QUANTIFIERS

12. QUANTIFIERS
QUANTIFIERS

13. QUANTIFIERS
QUANTIFIERS

15. THANK YOU!
THANK YOU!