4. [15 Pts] Let L = {(a,b) | a, b Z, (a - b) mod 3...

 

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4. [15 Pts] Let L = {(a,b) | a, b Z, (a - b) mod 3 = 0}. We want to program a robot that can get to each point (x, y) EL starting at (0,0). (a) [5 Pts] Give an inductive definition of L. This will describe the steps you want the robot to take to get to points in L starting at (0,0). Let L' be the set obtained by your inductive definition. (b) [5 Pts] Prove inductively that L'CL, i.e., every point that the robot can get to is in L. (c) Extra Credit [5 Pts] Prove that LC L', i.e., the robot can get to every point in L. To prove this, you need to give the path the robot would take to get to every point in L from (0,0), following the steps defined by your inductive rules. 4. [15 Pts] Let L = {(a,b) | a, b Z, (a - b) mod 3 = 0}. We want to program a robot that can get to each point (x, y) EL starting at (0,0). (a) [5 Pts] Give an inductive definition of L. This will describe the steps you want the robot to take to get to points in L starting at (0,0). Let L' be the set obtained by your inductive definition. (b) [5 Pts] Prove inductively that L'CL, i.e., every point that the robot can get to is in L. (c) Extra Credit [5 Pts] Prove that LC L', i.e., the robot can get to every point in L. To prove this, you need to give the path the robot would take to get to every point in L from (0,0), following the steps defined by your inductive rules.