Math135Lecture11StudentNotes

Math135Lecture11StudentNotes - Math 135 Lecture 11...

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Unformatted text preview: Math 135: Lecture 11: Congruence Definition 11.1 (Congruent) . Let m be a fixed positive integer. If a,b ∈ Z we say that a is congruent to b modulo m , and write a ≡ b (mod m ) if m | ( a- b ). If m- ( a- b ), we write a 6≡ b (mod m ). Example 11.2. Verify each of the following 1. 20 ≡ 2 (mod 6) 2. 2 ≡ 20 (mod 6) 3.- 20 ≡ 4 (mod 6) 4. 24 ≡ 0 (mod 6) 5. 5 6≡ 3 (mod 7) Equivalences a ≡ b (mod m ) ⇐⇒ m | ( a- b ) ⇐⇒ ∃ k ∈ Z 3 a- b = km ⇐⇒ ∃ k ∈ Z 3 a = km + b Congruence Behaves Like Equality Equality is an equivalence relation . That is, it has the following three properties: 1. reflexivity , 2. symmetry , 3. transitivity , Most relationships do not have these three properties. Example 11.3. • The relation greater than fails • The relation divides fails • The non-mathematical relation is a parent of fails Proposition 11.4 (Congruence Is An Equivalence Relation ( CER )) . Let a,b,c ∈ Z . Then 1. a ≡ a (mod m ). 2. If a ≡ b (mod m ), then b ≡ a (mod m ). 3. If a ≡ b (mod m ) and b ≡ c (mod m ), then a ≡ c (mod m ). 1 Math 135: Lecture 11: Congruence Recall: If a,b ∈ Z we say that a is congruent to b modulo m , and write a ≡ b (mod m ) if...
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Math135Lecture11StudentNotes - Math 135 Lecture 11...

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