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### 58 Cards in this Set

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 Proposition A sentence that is either true or false Conjunction The conjunction of P and Q, denoted P /\ Q, is true exactly when both P and Q are true. "P and Q." Disjunction The disjunction of P and Q, denoted P \/ Q, is true exactly when at least 1 is true. "P or Q" Negation The negation of P, denoted ~P, is true exactly when P is false. "Not P." Propositional Form A finite number of propositions together with logical connections Tautology A propositional form that is always true. Contradiction A propositional form that is always false. Denial The denial of a propositional form, P, is any propositional form equivalent to ~P. Two propositional forms P and Q are equivalent iff P and Q have both the same truth tables. What's a conditional sentence? Write out the truth table. A conditional sentence is "If P then Q" or P=>Q. P Q P=>Q T T T F T T T F F F F T In the statement P=>Q or "if P then Q," what is the consqeuent and what is the antecedent? P is the antecedent and Q is the consequent. Converse Let P and Q be propositions. The converse of P=>Q is Q=>P. Contropositive Let P and Q be propositions. The contropositive of P=>Q is ~Q=>~P. What's a biconditional sentence? Write out the truth table. For the propositions P and Q, the biconditional sentence P<=>Q is the proposition "P if and only if Q." P<=>Q is true when P and Q have the same truth tables. P Q P<=>Q T T T F T F T F F F F T Open Sentence/Predicate A sentece that contains one or more varables and which becomes a proposition only when the varaibles are replaced by the names of particular objects. Universe The objects under consideration (or objects used to replace the variables). Existential Quantifier For an open sentence P(x), the sentence (╣Ex)P(x) is read there exists x such that P(x) or for some x, P(x), and is true iff the truth set of P(x) is nonempty. Universal Quantifier For an open sentence P(x), the sentence (\-/x)P(x)is read for all x, P(x) and is true iff the truth set of P(x) is the entire universe. Even An integer z is even iff there exists a k in the integers such that z=2*k. Odd An integer z is odd iff there exists a k in the integers such that z=2*k+1 Divides Let a and b be integers. It's said that a divides b or a|b iff b=a*k for some k in the integers. Prime A natural number p is prime iff the only numbers that divide p are 1 and p. Absolute Value |x|={x if x>=0} {-x if x<0} Proof by Contropositive Assume ~Q : : ~P Therefore, ~Q=>~P Thus P=>Q by controposition. Set A set is a collection of objects. Use brackets {} when defining a set. Elements The objects in a given collection (or members of the set). The symbol is the C with the dash through it. Ex B={1,2,3,4} 1 є B. Empty Set Φ or {} is the set with no elements in it. Subset Let A and B be sets. If all the elements of A are within B, we say that A is a subset of B<=>(\-/x)(xεA=>xεB) When does set A equal set B? Set A and B are equal iff A(_C)B and B(_C)A. Define A(_C)B We say that for every x in y is also contained in B. Think of area which is the universe. An area B is contained within the universe. If A(_C)B, then an area A is contained within the area B. A(C_)B iff (\-/x)(xЄA=>xЄB) Power Set Let A be a set. The power set of A is the set whose elements are the subsets of A is and is denoted with a funky P. Union of A and B The set AUB is = {x|xЄA \/ xЄB} Intersection of A and B The set A∩B ={x:xЄA /\ xЄB} Difference of A and B The set A-B ={x:xЄA /\ xЄ~B} (except the mark goes on top of the B) Disjoint Two sets A and B are disjoint iff A∩B = Ø Complement If U is the universe and B(C_)U, then we define the complement of B to be the set of ~B = U-B. Universe The set of objects under consideration. Proof by Contradiction ` Prove P Pf: Suppose ~P : R : ~R (-><-) Therefore by contradiction, P Pairwise Disjoint An indexed collection of sets {Aα: α є Δ} is said to be pairwise disjoint iff for all α and β in Δ, if Aα ≠ Aβ, then Aα∩Aβ = Ø Principle of Mathematical Induction If a set S is a nonempty subset of lN with the following 2 properties: 1) 1єS 2) \\-/n є lN if n є S, then n+1 є S Then S=lN Inductive A set X is inductive iff X(C_)lN such that \\-/nєlN, if nєX, then (n+1)єX Well-Ordering Principle Every non-empty subset of lN has a least element. Cartesian Product Let A and B be sets. The sets of all ordered pairs having a first coordinate from A and a second coordinate from B is called the Cartestian Product of A and B is is written AxB. Thus AxB={(a,b):aєA/\bєB} Relation from A to B Let A and B be sets. R is a relation from A to B iff R is a subset of A x B. If (a,b)єR, we write aRb and say a is R-related (or just related) to b. Domain The domain of the relation R from A to B is the set Dom(R)={xєA:there exists yєB such that xRy) Range The range of the relationR is the set Rng(R)={yєB: there exist xєA such that xRy). Inverse If R is a relation from A to B, then the inverse of R is R^-1={(y,x):(x,y)єR}. Composite Let R be a relation from A to B, and let S be a relation from B to C. The composite of R and S is SoR={(a,c): there exists bєB such that (a,b)єR and (b,c)єS} function f is the relation from A to B such that i. Dom(f)=A ii. If f(x)=y and f(x)=z, then y=z Codomain What is the master codomain? The codomain is any set B such that Rng(f) _C B. The master codomain is ҐR. Equivalence Class If R is an equivalence relation from A to B x/R={yєB:xRy} Set of all equivalence classes If R is an equivalence relation on A, then A/R={x/R:xєA} Reflexive If R is a relation on A, then \-/xєA, xRx Symmetric If R is a relation on A, then \-/xєA and \-/yєA, if xRy, then yRx Transitive If R is a relation on A, then \-/xєA, \-/yєA, and \-/zєA, if xRy and yRz, then xRz. One-To-One A function is said to be 1-1 iff f(x)=f(y), then x=y. Onto A function is said to be onto B iff Rng(f)=B. Equivalence Relation R is a relation that is i. Reflexive ii. Symmetric iii. Transitive