Math 117 Exam 1
Necessary Condition
"∀x, r(x) is a ________ condition for s(x)" means "∀x, if ∼r(x) then ∼s(x)" or "∀x, if s(x) then r(x)"
Only If
"∀x, r(x) only if s(x)" means"∀x, if ∼s(x) then ∼r(x)" or "∀x, if r(x) then s(x)"
Associative Laws
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r) (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
Common Proof Mistakes
1) Argue from example in an instance when trying to prove "for all" 2) Using the same letter to represent different things 3) Jumping to conclusions 4) Begging the question (alluding to the proof's answer in the assumptions) 5) Misuse of "if" to generate uncertainty
Predicate
A sentence that contains a finite number of variables and becomes a statement when specific values are substituted for those variables. P(x): x is a mammal
Statement
A sentence that is true or false but not both - Evaluation does not change
Forward-Backwards Method
Listing assumptions and conclusions first when constructing a proof
Domain
Non-empty sets of values that may be finite or infinite and are substituted in the place of the predicate variable. - Can include numbers, sets of strings, sets of lists
Conjunction
Takes two statements and connects them - "and"/"or"
Contradiction
always false regardless of the truth values of the individual statements substituted for its statement variables.
Tautology
always true regardless of the truth values of the individual statements substituted for its statement variables.
p → q
if p is true then q is true
Quotient Remainder Theorem
n = dq +r and 0 ≤ r < d where n,d,q,and r ∈ Z
Prime
n ∈ Z is a prime if n > 1 and for all positive integers r and s, if n = r * s, then r = 1 or s = 1
Generalization
p ∴ p ∨ q q ∴ p ∨ q
Conjunction
p q ∴ p ∧ q
Converse Error
p → q q ∴p
Inverse Error
p → q ∼p ∴ ∼q (Invalid)
Transivity
p → q q → r ∴ p → r
Contrapositive
p → q ≡ ∼q → ∼p
Distributive Laws
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ≡ (p ∨q) ∧ (p ∨ r)
Indempotent Laws
p ∧ p ≡ p p ∨ p ≡ p
Specialization
p ∧ q ∴ p p ∧ q ∴ q
Commutative Laws
p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p
Identity Laws
p ∧ t ≡ p p ∨ c ≡ p
Absorption Laws
p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p
Proof by Division into Classes
p ∨ q p → r q → r ∴ r
Elimination
p ∨ q ∼q ∴ p p ∨ q ∼p ∴q
Universal Bound Laws
p ∨ t ≡ t p ∧ c ≡ c
Negation Laws
p ∨ ∼p ≡ t p ∧ ∼p ≡ c
Converse
q → p is the ____________ of p → q
Sufficient Condition
∀x, r(x) is a __________ condition for s(x)" means "∀x, if r(x) then s(x)
Negation
∼(p → q) ≡ ∼((∼p) ∨ q) or p ∧ ∼q - "for all" statements become logically equivalent to "some are not" (∼(∀x ∈ D, Q(x)) ≡ ∃x ∈ D such that ∼Q(x)) - "some are" statements become "all are not"
De Morgan's Laws
∼(p ∧ q) ≡ ∼p ∨ ∼q ∼(p ∨ q) ≡ ∼p ∧ ∼q
Double Negative Laws
∼(∼p) ≡ p
Contradiction
∼p → c ∴ p
Inverse
∼p → ∼q is the __________ of p → q
Negation of t and c
∼t ≡ c ∼c ≡ t where t = tautology and c = contradiction
