BSED MATH 8 Jamon.pdf Quantifiers Negation

Quantifiers

1. 88BSED MATHBSED MATH

2. LOGIC AND INCIDENCELOGIC AND INCIDENCE
GEOMETRYGEOMETRY
NEGATION ANDNEGATION AND
QUANTIFIERSQUANTIFIERS

3. LEARNING OBJECTIVESLEARNING 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.”

NEGATIONNEGATION

6. QUANTIFIERSQUANTIFIERS
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. QUANTIFIERSQUANTIFIERS
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.
QUANTIFIERSQUANTIFIERS

9. QUANTIFIERSQUANTIFIERS

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

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12. QUANTIFIERSQUANTIFIERS

13. QUANTIFIERSQUANTIFIERS

15. THANK YOU!THANK YOU!