REVIEW2

# REVIEW2 - contrapositive contradiction cases Be able to...

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THINGS TO KNOW FOR EXAM 2 1. Definitions (1) least element (2) well-ordered (3) principle of mathematical induction (4) minimum counterexample (5) relation (6) domain (7) codomain (8) range (9) reﬂexive (10) symmetric (11) transitive (12) equivalence relation (13) equivalence class (14) partition (15) a | b (16) congruence mod n (17) Division algorithm (18) Z n (19) well-deﬁned (20) function (21) image (22) mapping (23) equality of functions (24) one-to-one (25) injective (26) onto (27) surjective (28) bijective (29) identity function (30) composition (31) associative (32) inverse relation (33) permutation 2. Problems Be able to do all assigned homework prob- lems, and similar problems. 3. Theorems to know Be able to prove all of the theorems indi- cated below. Theorem 6.2 (pg. 130) Theorem 8.2 (pg. 181) Theorem 8.3 (pg. 182) Theorem 8.4 (pg. 183) Theorem 8.6 (pg. 185) Theorem 8.9 (pg. 191) Theorem 9.4 (pg. 203) Theorem 9.7 (pg. 208) Theorem 9.11 (pg. 209) 4. Skills you should have Be able to prove theorems by direct proof,
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Unformatted text preview: contrapositive, contradiction, cases. Be able to determine whether a statement is true or false, and give a proof or counterex-ample. Be able to prove or disprove facts about integers (even/odd), divisibility, congruence, real numbers, sets, rationality, irrationality. Be able to negate statements with quanti-ﬁers (possibly multiple quantiﬁers). Be able to compute in Z n . Be able to determine if a relation is an equivalence relation or a function. Be able to determine equivalence classes of an equivalence relation. Be able to tell if functions are injective, sur-jective, bijective. Be able to compute inverses of functions (when they exist). Be able to tell why a given function is or is not invertible. Be able to compose functions/permutations. Be able to prove theorems using induction (including theorems concerning sums, sequences, inequalities, congruences and divisibility) 1...
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