Proofs Flashcards
a mathematical object that is created for purposes
of simplification.
A Definition
a proposition or predicate assumed to be True.
An Axiom
a sequence of logic that demonstrates a Theory.
A Proof
▶ Even and Odd
▶ Prime and Composite
▶ Positive and Negative
▶ Addition, multiplication, subtraction, and division
▶ Closure
▶ Definition of Commutative and Associative
▶ And, Or, and Xor
These are examples of
Definitions
▶ Reflexive Property of equality “x = x”
▶ Symmetric Property of equality “x = y” ⇒ “y = x”
▶ 1 is the smallest natural number.
▶ Fundamental Theorem of Arithmetic
These are examples of
Axioms
4 main types of statements that will be discussed in class. (that aren’t axioms)
- Lemmas
- Theorems
- Corollaries
- Conjectures
“Easy” to prove statements; to be used by a
different statements.
Lemmas
“Hard” to prove statements; form the main point
of our argument.
Theorems
Statements that follow from a theorem.
Corollaries
Statements yet to be proved True.
Conjecture
Used when we need to show that whenever some
precondition is True, then some postcondition follows (A strong implication)
Direct Proof (Easy Proof)
What type of proof is this?
Direct Proof
Axiom of Even Oddness
Axiom of Even Oddness
How would one prove this theorem?
The ___ of an implication
Contrapositive
Contraposition
