# sol-01 - MATH 453 SOLUTIONS TO ASSIGNMENT 1 SEPTEMBER 8...

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M  453 S   A  1 S  8, 2004 Exercise 4 from Section 3, page 28 (a) Reﬂexivity, symmetry and transitivity for follow from the corresponding properties for equality, so is an equivalence relation. For instance, to check symmetry, we proceed as ( a 0 a 1 ) ( f ( a 0 ) = f ( a 1 )) ( f ( a 1 ) = f ( a 0 )) ( a 1 a 0 ). (b) Let A * = { [ a ] | a A } be the equivalence classes under . The map f : A B induces a map f * : A * B given by f * ([ a ]) = f ( a ). We will check that f * is a bijection. Given b B , there is an a A such that f ( a ) = b since f is surjective. Hence f * ([ a ]) = b and so f * is surjective. For injectivity, suppose f * ([ a 0 ]) = f * ([ a 1 ]). By the deﬁnition of f * we get f ( a 0 ) = f ( a 1 ), or a 0 a 1 . Hence [ a 0 ] = [ a 1 ] and f * is injective. ± Exercise 5 from Section 3, page 28 (a) Again we have to check the three conditions for an equivalence relation. For reﬂex- ivity, note that x - x = 0, so ( x , x ) S 0 . If y - x is an integer, then x - y = - ( y - x ) is too, and symmetry follows. Finally, if y - x and z - y are integers, then so is z - x = ( z - y ) + ( y - x ). Hence ( x , z ) S 0 and we have transitivity. If ( x , y ) S , then y = x + 1, or y - x = 1, an integer. Hence ( x , y ) S 0 . This shows that S 0 S . Given x R , the equivalence class [ x ] consists of all y R such that y - x Z . In other words, y is of the form x + n where n Z . Denoting the set { x + n | n Z } by x + Z , we have [ x ] = x + Z . Furthermore, given any x R , there is an x 0 [0 , 1) and an integer n such that x = x 0 + n . Hence [ x ] = [ x 0 ] for some x 0 [0 , 1). Also, no two distinct elements of [0 , 1) are S 0 -related, so R / S 0 = { x + Z | x [0 , 1) } . ± Exercise 5 from Section 5, page 39 Before looking at the speciﬁc subsets in this problem, let us consider an arbitrary subset A = { x | x satisﬁes some condition P } . Suppose P does not mix the indices, i.e. it is of the form “for all i , x i satisﬁes some condition C ( i ) that depends only on i ”. This means that the i th coordinate x i can only take values from the subsets A i of

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sol-01 - MATH 453 SOLUTIONS TO ASSIGNMENT 1 SEPTEMBER 8...

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