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-- May 2 In-Class Exercise Thread

 Given : X={1,2,3,4,5,6} and F={{1,2},{2,3},{4,5},{5,6},{1,6},{1,3,5}}
 
 
 Round 1: U = X; C = 0; S1 = {1,3,5}
 U = U - S = {2,4,6}
 C = C U {S} = {1,3,5}

 Round 2:  S2 = {1,2} 
 U = {4,6}
 C = {{1,3,5},{1,2}}
 
 
 Round 3:  S3 = {4,5} 
 U = {6}
 C = {{1,3,5},{1,2},{4,5}}
 
 
 Round 4: S4 = {5,6}
 U = 0
 C = {{1,3,5},{1,2},{4,5},{5,6}}
 
 
 |S3 - (S1 U S2)| vs |{5,6} - (S1 U S2)|
 {4,5} - {1,3,5,2} = {4}
 {5,6} - {1,3,5,2} = {6}
 |{4}| = |{6}| = 1 
 
 Hence, the inequality law is obeyed.

(Edited: 2018-05-05)

Given : X={1,2,3,4,5,6} and F={{1,2},{2,3},{4,5},{5,6},{1,6},{1,3,5}} Round 1: U = X; C = 0; S1 = {1,3,5} U = U - S = {2,4,6} C = C U {S} = {1,3,5} Round 2: S2 = {1,2} U = {4,6} C = {{1,3,5},{1,2}} Round 3: S3 = {4,5} U = {6} C = {{1,3,5},{1,2},{4,5}} Round 4: S4 = {5,6} U = 0 C = {{1,3,5},{1,2},{4,5},{5,6}} |S3 - (S1 U S2)| vs |{5,6} - (S1 U S2)| {4,5} - {1,3,5,2} = {4} {5,6} - {1,3,5,2} = {6} |{4}| = |{6}| = 1 Hence, the inequality law is obeyed.

-- May 2 In-Class Exercise Thread

 Step 1: S1 = {1,3,5} C = {{1,3,5}}
 Step 2: S2 = {1,2} C = {{1,3,5}, {1,2}}
 Step 3: S3 = {4,5} C = {{1,3,5}, {1,2}, {4,5}}
 Step 4: S4 = {5,6} C= {{1,3,5}, {1,2}, {4,5}, {5,6}}
 
 S3 - (S1 U S2) = {4}
 {5, 6} - (S1 U S2) = {6}
 |{6}| = |{4}| = 1
 yes

Step 1: S1 = {1,3,5} C = {{1,3,5}} Step 2: S2 = {1,2} C = {{1,3,5}, {1,2}} Step 3: S3 = {4,5} C = {{1,3,5}, {1,2}, {4,5}} Step 4: S4 = {5,6} C= {{1,3,5}, {1,2}, {4,5}, {5,6}} S3 - (S1 U S2) = {4} {5, 6} - (S1 U S2) = {6} |{6}| = |{4}| = 1 yes

-- May 2 In-Class Exercise Thread

  Starting with U = X; C = 0
  
  S1: {1,3,5},	C = {{1,3,5}}
  S2: {1,2}, 	C = {{1,3,5},{1,2}}
  S3: {4,5}, 	C = {{1,3,5},{1,2},{4,5}}
  S4: {5,6}, 	C = {{1,3,5},{1,2},{4,5},{5,6}}

  |S3 - (S1 U S2)| vs |{5,6} - (S1 U S2)|
  {4,5} - {1,2,3,5} = {4}
  {5,6} - {1,2,3,5} = {6}
  |{4}| = |{6}| = 1
  therefore, the inequality holds

Starting with U = X; C = 0 S1: {1,3,5}, C = {{1,3,5}} S2: {1,2}, C = {{1,3,5},{1,2}} S3: {4,5}, C = {{1,3,5},{1,2},{4,5}} S4: {5,6}, C = {{1,3,5},{1,2},{4,5},{5,6}} |S3 - (S1 U S2)| vs |{5,6} - (S1 U S2)| {4,5} - {1,2,3,5} = {4} {5,6} - {1,2,3,5} = {6} |{4}| = |{6}| = 1 therefore, the inequality holds

-- May 2 In-Class Exercise Thread

X={1,2,3,4,5,6}, F={{1,2},{2,3},{4,5},{5,6},{1,6},{1,3,5}}

Initially: U = X, C = 0.

Round 0: S1 = {1,3,5}, U = U - S = {2,4,6}, C = C U {S} = {1,3,5} Round 1: S2 = {1,2}, U = {4,6}, C = {{1,3,5},{1,2}} Round 2: S3 = {4,5}, U = {6}, C = {{1,3,5},{1,2},{4,5}} Round 3: S4 = {5,6}, U = 0, C = {{1,3,5},{1,2},{4,5},{5,6}}

|S3 - (S1 U S2)| vs |{5,6} - (S1 U S2)| {4,5} - {1,3,5,2} = {4} {5,6} - {1,3,5,2} = {6} |{4}| = |{6}| = 1

Therefore, the equality holds.

(Edited: 2018-05-09)

X={1,2,3,4,5,6}, F={{1,2},{2,3},{4,5},{5,6},{1,6},{1,3,5}} Initially: U = X, C = 0. Round 0: S1 = {1,3,5}, U = U - S = {2,4,6}, C = C U {S} = {1,3,5} Round 1: S2 = {1,2}, U = {4,6}, C = {{1,3,5},{1,2}} Round 2: S3 = {4,5}, U = {6}, C = {{1,3,5},{1,2},{4,5}} Round 3: S4 = {5,6}, U = 0, C = {{1,3,5},{1,2},{4,5},{5,6}} |S3 - (S1 U S2)| vs |{5,6} - (S1 U S2)| {4,5} - {1,3,5,2} = {4} {5,6} - {1,3,5,2} = {6} |{4}| = |{6}| = 1 Therefore, the equality holds.