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Problem One: Graphs and Logic, Problem Two: Relations and Functions and Problem Three: Functions

24/10/2020 Client: happyw Deadline: 7 Days

  


1. Consider a graph G = (V, E), where V = {A, B, C, D, E}. The edge set, E, is to be decided by you, according to criteria given below. The criteria are given in first-order logic, using the following functions and predicates:


Degree(v): a function giving the degree of vertex v.


Even(x): a predicate that is true if and only if the number x is even.


• > : a predicate with the usual mathematical greater-than meaning.


i. (2pts) Give a value for the edge set E that minimizes |E| and satisfies this criteria:


  ii. (2pts) Give a value for the edge set E that minimizes |E| and satisfies this criteria:


  


2. Given a function f : A A (where A is some set), we define a relation Rf  over A as


x Rf y if f(x) = y.


i. (2pts) Define a function f : that makes Rf  reflexive.


ii. (2pts) Define a function f : that makes Rf  symmetric, but neither reflexive nor transitive.


iii.  Define a function f : that makes Rf  transitive, but neither reflexive nor symmetric


3.   Given a function f : A B (where A and B are sets), we define a new function popf : ℘(A) → ℘(B) as follows:


Popf (S) = { y B | ∃x S. f(x) = y }


Notice that each input to popf  is a set S, because the domain of popf  is ℘(A) (so each S is some subset of A). Given some subset of A as input, popf  produces as output an element of ℘(B) (so each output is some subset of B), formed by applying f to each element of S.


i. (3pts) Let A = {0, 1} and B = {2, 3}. Define a function f : AB such that its resulting popf


function is not surjective.


f(x) =  2 if x is 0 2 if x is 1


ii. Let A and B be sets. Prove that if f : A B has a left inverse, then popf  has a left inverse. 

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