tut6large - { 1 , 2 , 3 }{ 1 , 2 , 3 } and draw its...

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Problems to Week 7 Tutorial — MACM 101 1. Determine which of the following relations R on the set A are reflexive, symmetric, transitive, and anti- symmetric. (a) A = { 1 , 2 , 3 } and R = { (1 , 1) , (1 , 2) , (2 , 1) , (2 , 2) } . Draw the graph and the matrix of this relation. (b) A is the set of all students at SFU, and ( x,y ) R means that the height of x differs from the hight of y by no more than one inch. (c) A is the set of ordered pairs of real numbers, that is, A = R × R , and (( x 1 ,x 2 ) , ( y 1 ,y 2 )) R if and only if x 1 = y 1 and x 2 y 2 . 2. Check that the following relations R on the set A are equivalence relations, find their equivalence classes, the number of equivalence classes, and determine which equivalence class the element z belongs to. (a) Let A be the set of all possible strings of 3 or 4 letters in alphabet { A,B,C,D } , let z = BCAD , and let ( x,y ) R if and only if x and y have the same first letter and the same third letter. (b) Let A be the power set of { 1 , 2 , 3 , 4 , 5 } , let z = { 1 , 2 , 3 } , and let ( x,y ) R if and only if x { 1 , 3 , 5 } = y ∩ { 1 , 3 , 5 } . 3. Prove that the following relation is an order on the Cartesian product
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Unformatted text preview: { 1 , 2 , 3 }{ 1 , 2 , 3 } and draw its di-agram: (( x 1 ,x 2 ) , ( y 1 ,y 2 )) R if and only if x 1 < x 2 , or x 1 = x 2 and y 1 y 2 . 1 (Such an order is called the lexicographic order .) 4. Prove that the following relation on the set of all nonempty subsets of { a,b,c,d } is an order, draw its diagram, nd all the maximal, minimal, least and greatest elements: ( x,y ) R if and only if x y. 5. Determine whether or not the following relations are functions. If a relation is a function, nd its range. (a) { ( x,y ) | x,y Z , y = x 2 + 7 } , a relation from Z to Z ; (b) { ( x,y ) | x,y R , y 2 = x } , a relation from R to R . 6. For each of the following functions, determine whether it is one-to-one and determine its range. (a) f : Z Z , f ( x ) = 2 x + 1; (b) f : Q Q , f ( x ) = 2 x + 1; (c) f : Z Z , f ( x ) = x 3-x ; 2...
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This note was uploaded on 12/12/2010 for the course MACM 201 taught by Professor Marnimishna during the Fall '09 term at Simon Fraser.

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tut6large - { 1 , 2 , 3 }{ 1 , 2 , 3 } and draw its...

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