MAT 300 Mathematical Structures
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each of the black spaces below before clicking
on it to display the answer.
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| Def: Tautology | Formulas that are always true
ex) P or not P
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| Def: Contradiction | Formulas that are always false
ex) P and not P
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| Def: Converse | The reversal of an if-then statement. Logically equivalent to inverse.
ex) P -> Q ; Q -> P
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| R | The set of all real numbers, containing all rational and irrational numbers. Q, N , Z are the subsets of the set R.
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| 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.
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| 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.
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| Z | Integers (positive or negatives including zero).
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| Def: Statement | An assertion (about mathematics) that is objectively either true or false
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| Def: Argument | A sequence of statements. All except the last are premises, while the last is a conclusion.
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| Def: Valid Argument | P1, P2, P3, ... , Pk | Q
Q is true when all Ps are true (conclusion is true whenever all premises are true
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| Def: Proof of an Assertion Q | A sequence of true statements connected with valid arguments and ending with Q
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| Def: Equivalence | Two statements with the same truth table
Symbol: <=>
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| Def: Set | A collection of objects (elements)
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| Free Variables | Truth value depends on variable
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| Bound Variables | Variable can be replaced or eliminated
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| DeMorgan's Law | ~(A or B) --> ~A and ~B
~(A and B) --> ~A or ~B
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| Def: Contrapositive | Reversing if-then and adding "not" to both sides (combines inverse and converse). Logically equivalent to conditional!
ex) P --> Q ; ~Q --> ~P
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| Def: Inverse | Adds not to both sides. Logically equivalent to converse.
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| Def: Power Set | The set whose elements are all subsets of A
P(A) = { x | x is a subset of A }
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| Commutative Laws | P ^ Q is equivalent to Q ^ P
P or Q is equivalent to Q or P
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| 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
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| Idempotent Laws | P and/or P is equivalent to P
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| 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)
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| Absorption Laws | P or (P ^ Q) is equivalent to P
P ^ (P or Q) is equivalent to P
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| Modus Ponens | If you know that P is true and P --> Q is true, then Q is also true
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| Modus Tollens | If you know that P --> Q is true and Q is false, then Q is false
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| Proof by Contradiction | Show that if a proposition is false, a contradiction is implied
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| Def: Elements | Objects in sets
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