Algebra 2 Notes 1 Flashcards
(1D-N1-1.1) Let a, b ∈ Z. Then a divides b if …
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(1P-N1-1.2) (i) If a|b and a|c then a|bd + ce, (ii) If a|b and b|c then a|c. (iii) if a, b ̸= 0 and a|b and b|a then a = ±b.
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(1D-N1-1.3) Factorisation: trivial, composite. Prime, units.
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(1T-N1-1.4) Division theorem
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(1D-N2-1.5) Highest Common Factor, Coprime
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(1T-N2-1.6) Euclid’s Algorithm
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(1D-N3-1.7) Linear Combinations
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(1T-N3-1.8) Let a, b be non-zero integers and x an integer. Then x is a linear combination of a and b if and only if hcf(a,b) | x.
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(1L-N3-1.9) the h-k Lemma’
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(1P-N3-1.10) Let p be a prime number and a and b be integers. Then p|ab ⇒ p|a or or p|b
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(1C-N4-1.11) Let p be a prime number and a1, a2, …, an be integers. Then if p divides a1a2…an, then p divides at least one of the ai.
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(1T-N4-1.12) Unique factorisation (in Z 2014)
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(1T-N5-1.14) Infinitely Many primes
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(1D-N6-1.15) Given R be a relation on a set X. Then Reflexive, Symmetric, Transitive
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(1D-N6-1.16) Equivalence Relation
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(1E-N6-1.17) Show R be defined on Z by xRy if y−x is a multiple of 3 is an ER. Then later different EC’s, Partitions
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(1D-N6-1.18) Equivalence Class
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(1P-N6-1.19) Let R be an equivalence relation on a set X. Then (i) [a] = [b] ⇔ aRb, (ii) [a] ∩ [b] = ∅ or [a] = [b]
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(1D-N7-1.20) A partition of a set X
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(1T-N7-1.21) Let R be an equivalence relation on a set X. Then the distinct equivalence classes form a partition of X, i.e
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(1R-N7-) Key notation
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(1T-N8-1.22) Let n be an integer ≥ 2. Then the relation on Z defined by aRb if b − a is a multiple of n is an equivalence relation. A set of representatives is given by 0,1,2,…n−1, i.e. Let Zn denote the set of equivalence classes. Then the operations defined by
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