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Logic and Proof

Logic and Proof

Proofs Exercises

Example 1

What is the reason for statement 7?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = C?

Example 2

What is the reason for statement 8?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0?

Example 3

What is the reason for statement 9?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0 

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BB?

Example 4

What is statement 11 given its reason?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BBMultiplication Property (1)
10. AB = CBSubstitution Property (9 and 2)
11. ?Division Property (10 and 6)

Example 5

What is statement 12 given its reason?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BBMultiplication Property (1)
10. AB = CBSubstitution Property (9 and 2)
11. Division Property (10 and 6)
12. ?Substitution Property (11 and 5)

Example 6

What is statement 13 given its reason?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BBMultiplication Property (1)
10. AB = CBSubstitution Property (9 and 2)
11. Division Property (10 and 6)
12.Substitution Property (11 and 5)
13. ?Multiplication Property (12)

Example 7

What is the reason for statement 14?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove:

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BBMultiplication Property (1)
10. AB = CBSubstitution Property (9 and 2)
11. Division Property (10 and 6)
12. Substitution Property (11 and 5)
13. Multiplication Property (12)
14. X = Z?

Example 8

What is the reason for statement 15?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove:

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BBMultiplication Property (1)
10. AB = CBSubstitution Property (9 and 2)
11. Division Property (10 and 6)
12. Substitution Property (11 and 5)
13. Multiplication Property (12)
14. X = ZTransitive Property (3 and 4)
15. ?
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