Math 3000 Running Glossary


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1 Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set ( ): A set containing no elements. 3. Natural Numbers (N): {1, 2, 3,...} 4. Integers (Z): {..., 3, 2, 1, 0, 1, 2, 3,...} or {x : x N} { x : x N} {0} 5. Rational Numbers (Q): { m n : m Z and n Z where n 0 } 6. Real Numbers (R): Any number that can be written in decimal form. (note that this is not the precise definition, but it is informally correct) 7. Complex Numbers (C): {a + bi : a R and b R}. Note that it is understood there that i = Cardinality of a set S (denoted S ): The number of elements in S. 9. A B: A is a subset of B if every element of A is also an element of B. 10. Power Set of a set A (P(A)): P(A) = {S : S is a subset of A} (Note that the empty set and the set A are both members of the power set) 11. Union A B: A B = {x : x A or x B} 12. Intersection A B: A B = {x : x A and x B} 13. Disjoint Sets: If two sets A and B have no elements in common, then A B = and A and B are said to be disjoint. 1
2 14. Set Difference A B or A\B: A B = A\B = {x : x A and x / B} 15. Complement Ā: Ā = {x : x U and x / A} (U is the universal set) 16. Cartesian Product: A B = {(a, b) : a A and b B} Chapter 2: 17. Statement: A statement is a declarative sentence or assertion that is true or false (but not both). 18. Open Sentence: An open sentence is a declarative sentence that contains one or more variables. 19. Disjunction: (be able to give a truth table) The statement P or Q. Also denoted P Q 20. Conjunction: (be able to give a truth table) The statement P and Q. Also denoted P Q 21. Conditional Statement: (be able to give a truth table) The statement: If P, then Q. Also denoted P = Q. 22. Converse: The converse of P = Q is Q = P 23. The Biconditional: (be able to give a truth table) The statement: P = Q along with its converse Q = P. Another notation: (P Q). Another notation: (P if and only if Q). 24. Tautology: A statement is a tautology if it is true for all combinations of truth values. 25. Logical Equivalence: When two statements have the same truth values for all combinations of truth values. 26. The quantifier: for all 27. The quantifier: there exists Chapter 3: 28. Axiom: A true mathematical statement whose truth is accepted without proof. 2
3 29. Theorem: A true mathematical statement whose truth can be verified. 30. Corollary: A mathematical result that can be deduced from some earlier result. 31. Lemma: A mathematical result that is useful in establishing the truth of some other result. 32. Trivial Proof: If Q(x) is true for all x S (regardless of the truth of P (x)), then P (x) = Q(x) is true trivially. 33. Vacuous Proof: If P (x) is false for all x S (regardless of the truth of Q(x)), then P (x) = Q(x) is true vacuously. 34. Direct Proof: To prove P (x) = Q(x) with a direct proof, assume that P (x) is true and prove that Q(x) is true. 35. Contrapositive: The contrapositive of P = Q is Q = P and is logically equivalent to P = Q. 36. Proof by contrapositive: To prove P (x) = Q(x) with a proof by contrapositive, assume that Q(x) is false and prove that P (x) is false. 37. An even integer: An integer n is even if there exists an integer k such that n = 2k. 38. An odd integer: An integer n is odd if there exists an integer k such that n = 2k Without Loss of Generality (WLOG): This phrase indicates that the proofs of two situations are identical, so the proof of only one of these is needed. Chapter 4: 40. a divides b. Denoted: a b (technically a characterization) For integers a and b with a 0, we say that a divides b if there is an integer c such that b = ac. 41. b is a multiple of a: If a b, then we say that b is a multiple of a. 42. a is a divisor of b: If a b, then we say that a is a divisor of b. 43. a does not divide b. Denoted (a b): There is not an integer c such that b = ac. 3
4 44. a is congruent to b modulo n. Denoted a b (mod n): a b (mod n) if n (a b). Think of this as: a and b have the same remainder when divided by n. 45. Absolute Value: 46. Triangle Inequality: x + y x + y x = { x if x 0 x if x < Fundamental Properties of Set Operations (this is technically a theorem) (a) Commutative Laws i. A B = B A ii. A B = B A (b) Associative Laws i. A (B C) = (A B) C ii. A (B C) = (A B) C (c) Distributive Laws i. A (B C) = (A B) (A C) ii. A (B C) = (A B) (A C) (d) DeMorgan s Laws i. A B = A B ii. A B = A B Chapter 5 (There aren t many new definitions in chapter 5, but some very powerful proof techniques are introduced.) 48. Counterexample: An element y S is called a counterexample of the statement ( x S, R(x)) if R(y) is false. Chapter Well Ordered Set A set, S, is called well ordered if every nonempty subset of S has a least element. (An example is the natural numbers (see below). An example of a set that is not well ordered is the set of complex numbers.) 50. Well Ordering Priciple The set of natural numbers (N) is well ordered. (This is really an axiom) 51. The Principle of Mathematical Induction (theorem 6.1): For each positive integer n, let P (n) be a statement. If 4
5 (1) P (1) is true and (2) the implication If P (k), then P (k + 1) is true for every positive integer k, then P (n) is true for every positive integer n. Chapter 8 (We only cover sections ) 52. Relation A relation, R, from A to B is a subset of A B. In other words, R is a set of ordered pairs, where the first coordinate of the pair belongs to A and the second coordinate belongs to B. Notation: arb reads a is related to b. 53. Domain of a relation dom(r) = {a A : (a, b) R for some b B} 54. Range of a relation ran(r) = {b B : (a, b) R for some a A} 55. Relation on a set A A relation on a set A is a relation from A to A. In other words, a relation on set A is a subset of A A. 56. Reflexive property A relation is reflexive if xrx for every x in the set A. 57. Symmetric property A relation R defined on a set A is symmetric if whenever xry, then yrx for all x, y A 58. Transitive property A relation R defined on a set A is transitive if whenever xry and yrz, then xrz for all x, y, z A. 59. Distance The distance between two real numbers a and b is a b. 60. Equivalence Relation A relation R on a set A is called an equivalence relation if R is reflexive, symmetric, and transitive. 5
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