INDEX: Back to "examples" page Symbols and Truth Tables Protocol Expressions and Tautologies Quantifiers Glossary How to do a Proof Link	Foundations of Higher Mathematics
Bi-Conditional - It will be the second variable if and only if the first one is true.	Propositions: Propositions are statements that are either true (1) or false (0). Truth Tables    Conditional propositions - are stated "if P. then Q". P is the hypothesis or conclusion and Q is the conclusion.

1.2 Expressions and Tautologies

Protocol - The order of things: Propositional Expression - While a proposition is either t or f, a expression has no meaning until its variables are replaced by propositions. An expression cannot be t or f. expressions have truth tables and propositions have none. EXAMPLE: [SYMBOL KEY] Logically Equivalent - Expressions are said to be equivalent if all the truth tables are the same when all the variables are tried with t or f. EXAMPLE: When X and Y are equalivant we write. [SYMBOL KEY] *cannot be used in a proposition EXAMPLE: [SYMBOL KEY] EXAMPLE: [SYMBOL KEY] Tautology - An expression that equals true no matter what is used for its variables. EXAMPLE: [SYMBOL KEY] IMPORTANT:[SYMBOL KEY] Part a) means that in order for ? Part d) and e) are called de Morgan's laws   Contradiction - An expression that equals false no matter what is used for its variables. EXAMPLE: [SYMBOL KEY] Contrapositive - To be contrapositive you must flip and negate. EXAMPLE: [SYMBOL KEY] Converse - To be converse you must flip. EXAMPLE: [SYMBOL KEY]

1.3 QUANTIFIERS

Signs Constants - a member of our given universe. Variables - x<4 has a variable and thus is not a proposition. Propositional function or open sentance - if variable is replaced by a specific real number (called a universe or set of meanings). Apparent or Bound Variable - is a variable in a true sentance. EXAMPLE: x, x+2=2+x Actual or Free Variable - is a variable that has to be replaced by a constant. EXAMPLE: x+3=7 Truth Set - The colection of objects that can replace the variable. EXAMPLE: in x<4 the truth set is all real numbers that are less than 4. Existential Quantifiers for some x there exists there are x and y EXAMPLE: There exists x such that P(x) - is written [SYMBOL KEY] Universal Quantifiers for each x for any x for all x EXAMPLE: For all x, P(x) - is written [SYMBOL KEY] Proofs with Quantifiers Must begin proofs with a negation. EXAMPLE:[SYMBOL KEY] Counterexample Object t in the set of meanings such that P(t) is false Divides m divides n if there is an integer q such that n = m x q. m = divisor n = divisible by m n = multiple of m Primes Natural numbers whos only divisable by themselves. Perfect Square n is a perfect square if there is a integer k such that n = k^2.

GLOSSARY

Bi-Conditional Conditional propositions (as stated) Constants Contradiction Contrapositive Converse Counterexample Divides de Morgan's laws Existential Quantifiers Logically Equivalent Perfect Square Primes Proofs with Quantifiers Propositions Propositional Expression Propostitional Function Signs Tautology Truth Set Universe (Set of Meanings) Universal Quantifiers Variables

HOW TO DO A PROOF

Start with a theorem Write dow hypothesis use tricks and make notes use algebra if necessary and note it other tricks are axioms what is good for the goose is good for the gander Now write down the conclusion (should be on the side of the hypothesis as it is in the scratch section of extra notes 3) use tricks on it also Now the proof - just read doun the lists and transcribe to perfect sentance structure. Theorem: Let a, b, and c be real numbers. If a<b and b<c, then a<c. Hypothesis: a<b and b<c b-a is positive and c-b is positive (definition of <) (b-a)+(c-b) = c-a (algebra) (b-a)+(c-b) is positive (axiom 1) c-a is positive Conclusion: a<c c-a is positive (definition of <) PROOF: Suppose a<b and b<c. Then b-a is positive and c-b is positive. By axiom 1, (b-a)+(c-b) is positive, and (b-a)+(c-b)=c-a. Therefore c-a is positive and thus a<c.