a3s - CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE...

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Unformatted text preview: CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING COMP238 MATHEMATICS FOR COMPUTER SCIENCE I ASSIGNMENT 3 SOLUTIONS WINTER 2009 1. If A and B are sets and f : A- B then for any subset S of B the pre-image of S is f- 1 ( S ) { a A : f ( a ) S } . Note that f- 1 ( S ) is well defined even if f does not have an inverse! Let S and T be subsets of B . Prove that f- 1 ( S T ) = f- 1 ( S ) f- 1 ( T ) . PROOF: x f- 1 ( S T ) f ( x ) S T f ( x ) S or f ( x ) T x f- 1 ( S ) or x f- 1 ( T ) x f- 1 ( S ) f- 1 ( T ) . Prove that f- 1 ( S T ) = f- 1 ( S ) f- 1 ( T ) . PROOF: x f- 1 ( S T ) f ( x ) S T f ( x ) S and f ( x ) T x f- 1 ( S ) and x f- 1 ( T ) x f- 1 ( S ) f- 1 ( T ) . 2. Show that the function g : ( R- { 1 } )- ( R- { 1 } ) given by g ( x ) = x + 1 x- 1 , is a bijection ( i.e. , one-to-one and onto). SOLUTION: A function is one-to-one and onto if and only if it is invertible. Thus it suffices to show that g has an inverse. In fact g- 1 ( x ) = g ( x ), because for any x R-{ 1 } we have g ( g ( x )) = g ( x + 1 x- 1 ) = x +1 x- 1 + 1 x +1 x- 1- 1 = ( x + 1) + ( x- 1) ( x + 1)- ( x- 1) = x . 1 3. When an integer b is divided by 12, the remainder is 5. What is the remainder when 8 b is divided by 12?...
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a3s - CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE...

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