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Unformatted text preview: 1.1 – Intro: Statement is true or false, not both. DeMorgan’s Theorem : ~(p ∨ q) == ~p ^ ~q ~(p^q) == ~p ∨ ∼ q. Tautology : All Concluding values of TT=True Contradiction : Same as above but TT=False In general, questions don’t = statements. 1.2 – If Statements : If p, then q == p ﾆ q == ~p ∨ q (p:q:Conclusion) ﾆ (TTT,TFF,FTT,FFT) Contrapositive ~q ﾆ ~p Converse : q ﾆ p Inverse: ~p ﾆ ~q p only if q : If not q, then not p : p ﾆ q means ~p ∨ q p if q : If q, then p: q ﾆ p. Biconditional : p iff q: p must have same TT as q, else false. P ↔ Q (p:q:Conclusion) ﾆ (TTT,TFF,FTF,FFT) *** p ↔ q (p ﾆ q) ^ (q ﾆ p) Sufficient Condition : p is a necessary condition for q: If p, then q == p ﾆ q. Necessary Condition : p is a sufficient condition for q: If ~p, then ~q === ~p ﾆ ~q === q ﾆ p Negating : ~(p ﾆ q) == ~(~p ∨ q) == (p ^ ~q) Notes: p ﾆ q ≠ q ﾆ p, p ﾆ q ≠ ~p ﾆ ~q, p ﾆ q == ~q ﾆ ~p 1.3 – Valid Forms: Argument: Sequence of statements, called premises. There is a final statement called a conclusion. Argument is of valid form when: 1.) In TT, find all critical rows (row where all premises = true). 2.) If all conclusions = True for all critical rows, statement is of valid form, else not in valid form. Modus Ponens : p ﾆ q, p, ∴ q. Modus Tollens : p ﾆ q, ~q, ∴ ~p. Disjunctive Addition: p, ∴ p ∨ q or q, ∴ p ∨ q. Disjunctive Syllogism : p ∨ q, ~q, ∴ p or p ∨ q, ~p, ∴ q. Conjunctive Addition: p, q , ∴ p ∨ q. Conjunctive Simplification: p ^ q, ∴ p or p ^ q, ∴ q. Hypothetical Syllogism: p ﾆ q, q ﾆ r, ∴ p ﾆ r. Division into cases: p ∨ q, p ﾆ r, q ﾆ r, r. Rule of Contradiction : ~p ﾆ c, ∴ p. 5 – Sets : If x is a member of Set S, x ∈ S, else x not in S. S = {a, b, c}: Set s consists exactly of the elements a, b, c. Order doesn’t matter and ignore repetition. i.e {b,a,c} and {a,a,b,b,c,c}. Use (…) to indicate pattern continues. T = {x ∈ S  (P(x))} == T ‘equals’ the ‘set of x’ in S, s.t P(x). Set of all permissible objects = The Universe = U, {x ∈ U  (P(x))}. Φ = Empty Set A = Subset of B , A C B (B contains A), iff every element in A is also in B. (B has A’s stuff) A = B iff A has same exact elements in B. A = Proper Subset of B ( A C B) iff (A C B) and B not equal to A. Ex: A = {0,1,2,3] B = {1,2} B C A but B does not equal A. Intersection of A and B : A ∩ B = Elements which are both in A and B. Union of A and B : A U = Elements which are either in A or B. Example: A = {0,1,2,3} B = {0,1,2,4}. A U B = {0,1,2,3,4.}. A ∩ B = {0,1,2} Difference between A and B = B – A: Elements in B that are not in A. Compliment : A C : All elements in Set U which are not in A....
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This note was uploaded on 04/17/2008 for the course MTH 314 taught by Professor Forgot during the Spring '08 term at Ryerson.
 Spring '08
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