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# Exam 1

### MAT 300 Mathematical Structures

Def: Tautology Formulas that are always true ex) P or not P
Def: Contradiction Formulas that are always false ex) P and not P
Def: Converse The reversal of an if-then statement. Logically equivalent to inverse. ex) P -> Q ; Q -> P
R The set of all real numbers, containing all rational and irrational numbers. Q, N , Z are the subsets of the set R.
Q The set of all rational numbers. The rational numbers are any number which could be written like p/q, where p and q are integers.
N The set of all natural numbers like:1,2,3,4,5,....... The set has starting number 1 and the consecutive numbers increments by 1 . It has no end.
Z Integers (positive or negatives including zero).
Def: Statement An assertion (about mathematics) that is objectively either true or false
Def: Argument A sequence of statements. All except the last are premises, while the last is a conclusion.
Def: Valid Argument P1, P2, P3, ... , Pk | Q Q is true when all Ps are true (conclusion is true whenever all premises are true
Def: Proof of an Assertion Q A sequence of true statements connected with valid arguments and ending with Q
Def: Equivalence Two statements with the same truth table Symbol: <=>
Def: Set A collection of objects (elements)
Free Variables Truth value depends on variable
Bound Variables Variable can be replaced or eliminated
DeMorgan's Law ~(A or B) --> ~A and ~B ~(A and B) --> ~A or ~B
Def: Contrapositive Reversing if-then and adding "not" to both sides (combines inverse and converse). Logically equivalent to conditional! ex) P --> Q ; ~Q --> ~P
Def: Inverse Adds not to both sides. Logically equivalent to converse.
Def: Power Set The set whose elements are all subsets of A P(A) = { x | x is a subset of A }
Commutative Laws P ^ Q is equivalent to Q ^ P P or Q is equivalent to Q or P
Associative Laws When a statement has all "and"s or "or"s, it doesn't matter how you separate them P ^ (Q ^ R) is equivalent to (P ^ Q) ^ R P or (Q or R) is equivalent to (P or Q) or R
Idempotent Laws P and/or P is equivalent to P
Distributive Laws P ^ (Q or R) is equivalent to (P ^ Q) or (P ^ R) P or (Q ^ R) is equivalent to (P or Q) ^ (P or R)
Absorption Laws P or (P ^ Q) is equivalent to P P ^ (P or Q) is equivalent to P
Modus Ponens If you know that P is true and P --> Q is true, then Q is also true
Modus Tollens If you know that P --> Q is true and Q is false, then Q is false
Proof by Contradiction Show that if a proposition is false, a contradiction is implied
Def: Elements Objects in sets
Created by: arianaflores