Introduction to Abstract Mathematics

Logic: Deciding whether or not a deduction is valid

Possible deduction: Hypotheses: ① cows are green
(valid deduction)
② Cows are not g


Conclusion: Cows are the sam


Assertion: a sentence that is either true or fals
(has a distinct "truth value")


Non-Examples: Comments:
• Is this abstract algebra?
• Assertions do n
• Pink is the best color. • Assume no gr
• Shucks!

Examples: Deduction: a s

• The sky is blue follow

• A cow has 4 legs. (all h
• Pi is equal to 3. mu




*
Deduction Ex: Valid or not?
a) yp: The earth is round.

yp: The earth is green.
H

,Logic Puzzles
knights & knaves:

0 Everyone is a knight or a Knave (o
② Everyone knows the status of ev

③ knights tell the truth. Knaves
④ Spys can do either.

Ex: You meet Rick & Morty.
Rick says, "hey, we're knights."
Morty say," Geez, Rick, you're a K
what are they?

Rick: Knave Morty: Kni

Case 1: Rick is a knight. Ca
• Morty is also a knight b/c • R
Rick always tells the truth.
•M
• Morty tells the truth, but
w
stated that Rick is a Knave.
(Contradiction)
- Rick cannot be a knight.

,Symbilizing Statements

Propositional Logic: Convenient way to expres

rogical relations. Assertions are represented
w/capital letters

Ex. Hyp: (i) If it's Friday, we don't have c
(2) Today is Friday.

Cont: We don't have class tomorrow

Hyp:(1)If B, men C Notation
- "therefore"
(2) B

Conel:c



Logical Connectives

i.e. ∀ E, 0 ∃ 870 704/✗-a/ < S ⇒

Symbol Nickname Meaning
It is not the
"Not"

, Truth Values of Assertions

EX. Find the truth value of
(AAB)⇒ (Cv-B) when:

a) Ast, B = F, CI
(Arb) ⇒ (Cv-B)
F ⇒ T (start w/F, vacuously true)
T


b) A=T, B=T, C=F
(AAB) ⇒ (CWB)
T⇒ F
F