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Data Structures &amp; Alogs HW_Part_2

# Data Structures &amp; Alogs HW_Part_2 - 2 Mathematical...

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2 Mathematical Preliminaries 2.1 (a) Not re fl exive if the set has any members. One could argue it is sym- metric, antisymmetric, and transitive, since no element violate any of the rules. (b) Not re fl exive (for any female). Not symmetric (consider a brother and sister). Not antisymmetric (consider two brothers). Transitive (for any 3 brothers). (c) Not re fl exive. Not symmetric, and is antisymmetric. Not transitive (only goes one level). (d) Not re fl exive (for nearly all numbers). Symmetric since a + b = b + a , so not antisymmetric. Transitive, but vacuously so (there can be no distinct a , b , and c where aRb and bRc ). (e) Re fl exive. Symmetric, so not antisymmetric. Transitive (but sort of vacuous). (f) Re fl exive – check all the cases. Since it is only true when x = y , it is technically symmetric and antisymmetric, but rather vacuous. Like- wise, it is technically transitive, but vacuous. 2.2 In general, prove that something is an equivalence relation by proving that it is re fl exive, symmetric, and transitive. (a) This is an equivalence that effectively splits the integers into odd and even sets. It is re fl exive ( x + x is even for any integer x ), symmetric (since x + y = y + x ) and transitive (since you are always adding two odd or even numbers for any satisfactory a , b , and c ). (b) This is not an equivalence. To begin with, it is not re fl exive for any integer. (c) This is an equivalence that divides the non-zero rational numbers into positive and negative. It is re fl exive since x ˙ x > 0 . It is symmetric since x ˙ y = y ˙ x . It is transitive since any two members of the given class satisfy the relationship.

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Data Structures &amp; Alogs HW_Part_2 - 2 Mathematical...

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