Logic and Proof Exercises

Example 1

What is the reason for statement 7?

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

Prove:

Statements	Reasons
1. A = B	Given
2. B = C	Given
3. X = Y	Given
4. Y = Z	Given
5. H = G	Given
6. G ≠ 0	Given
7. A = C	?

Hint

One of the very first properties we learned.

Example 2

What is the reason for statement 8?

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

Prove:

Statements	Reasons
1. A = B	Given
2. B = C	Given
3. X = Y	Given
4. Y = Z	Given
5. H = G	Given
6. G ≠ 0	Given
7. A = C	Transitive Property (1 and 2)
8. H ≠ 0	?

Hint

Use Statements 5 and 6.

Example 3

What is the reason for statement 9?

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

Prove:

Statements	Reasons
1. A = B	Given
2. B = C	Given
3. X = Y	Given
4. Y = Z	Given
5. H = G	Given
6. G ≠ 0	Given
7. A = C	Transitive Property (1 and 2)
8. H ≠ 0	Substitution Property (5 and 6)
9. AB = BB	?

Hint

What is happening to the first statement?

Example 4

What is statement 11 given its reason?

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

Prove:

Statements	Reasons
1. A = B	Given
2. B = C	Given
3. X = Y	Given
4. Y = Z	Given
5. H = G	Given
6. G ≠ 0	Given
7. A = C	Transitive Property (1 and 2)
8. H ≠ 0	Substitution Property (5 and 6)
9. AB = BB	Multiplication Property (1)
10. AB = CB	Substitution Property (9 and 2)
11. ?	Division Property (10 and 6)

Hint

What are we dividing by?

Example 5

What is statement 12 given its reason?

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

Prove:

Statements	Reasons
1. A = B	Given
2. B = C	Given
3. X = Y	Given
4. Y = Z	Given
5. H = G	Given
6. G ≠ 0	Given
7. A = C	Transitive Property (1 and 2)
8. H ≠ 0	Substitution Property (5 and 6)
9. AB = BB	Multiplication Property (1)
10. AB = CB	Substitution Property (9 and 2)
11.	Division Property (10 and 6)
12. ?	Substitution Property (11 and 5)

Hint

We can do this in part because of statement 8.

Example 6

What is statement 13 given its reason?

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

Prove:

Statements	Reasons
1. A = B	Given
2. B = C	Given
3. X = Y	Given
4. Y = Z	Given
5. H = G	Given
6. G ≠ 0	Given
7. A = C	Transitive Property (1 and 2)
8. H ≠ 0	Substitution Property (5 and 6)
9. AB = BB	Multiplication Property (1)
10. AB = CB	Substitution Property (9 and 2)
11.	Division Property (10 and 6)
12.	Substitution Property (11 and 5)
13. ?	Multiplication Property (12)

Hint

What should we multiply by? Well, what are we trying to prove?

Example 7

What is the reason for statement 14?

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

Prove:

Statements	Reasons
1. A = B	Given
2. B = C	Given
3. X = Y	Given
4. Y = Z	Given
5. H = G	Given
6. G ≠ 0	Given
7. A = C	Transitive Property (1 and 2)
8. H ≠ 0	Substitution Property (5 and 6)
9. AB = BB	Multiplication Property (1)
10. AB = CB	Substitution Property (9 and 2)
11.	Division Property (10 and 6)
12.	Substitution Property (11 and 5)
13.	Multiplication Property (12)
14. X = Z	?

Hint

Haven't we covered this already?

Example 8

What is the reason for statement 15?

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

Prove:

Statements	Reasons
1. A = B	Given
2. B = C	Given
3. X = Y	Given
4. Y = Z	Given
5. H = G	Given
6. G ≠ 0	Given
7. A = C	Transitive Property (1 and 2)
8. H ≠ 0	Substitution Property (5 and 6)
9. AB = BB	Multiplication Property (1)
10. AB = CB	Substitution Property (9 and 2)
11.	Division Property (10 and 6)
12.	Substitution Property (11 and 5)
13.	Multiplication Property (12)
14. X = Z	Transitive Property (3 and 4)
15.	?

Hint

Only one step remains. What did you multiply in statement 13?

Logic and Proof Exercises

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

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