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Assignment - Homework 4

Assignment - Homework 4 - x R y iﬀ b x 3 cb y 3 c What...

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Homework 4 MA 2312 Due Tuesday at the beginning of your recitation 1. Precisely define (a) A relation on the reals where two numbers relate if they are relatively prime. (b) A function that takes a set of integers and returns the sum of all elements in the set. (c) A relation on the sets of integers where two sets relate if they share at least one element. 2. Consider the following relations: (a) R 1 R × R ; R 1 = { ( x, y ) |b x c = d y e} (b) R 2 ( R × R ) × ( R × R ); R 2 = { (( x, y ) , ( a, b )) | ( b x c = b a c ) ( b y c = b b c ) } (c) Let W = the set of all English words in the Oxford English Dictionary (OED). R 3 W × W ; R 3 = { ( x, y ) | x does not come after y in the OED } Which of these are reflexive, symmetric, transitive, and antisymmetric? Which are partial orders? Which are equivalence relations? 3. Let R be an equivalence relation on Z + ∪ {

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Unformatted text preview: x R y iﬀ b x 3 cb y 3 c . What is . . . (a) 5 R 7 (b) [6] R (c) [10] R = [11] R (d) (2 , 4) ∈ R 4. Consider the function f : R 7→ R , where f ( x ) = 1 + sin x . (a) What it the domain of f ? (b) What is the codomain of f ? 1 (c) What is the range of f ? (d) Is f one-to-one? (e) Is f onto? (f) Is f invertible? If not, deﬁne a new function f in the same spirit as f , but that is invertible? 5. Deﬁne a one-to-one and onto function g : Z 7→ Z + . 6. Consider two relations A and B on a set S . • If A and B are reﬂexive, must A ◦ B be reﬂexive? Prove your answer. • If A and B are symmetric, must A ◦ B be symmetric? Prove your answer. 2...
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