a relation matrix linking survey stations to control points Given rows (elements of the first set) (n_r) = 7.0000; columns (elements of the second set) (n_c) = 10.0000, determine the matrix entries (E) in entries.

Start with the thinking

• The governing relation printed in this handbook section is Entries in a relation (adjacency) matrix.
• Everything except E is given, so isolate E symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A relation between two finite sets is stored as a Boolean matrix of relation, so the storage cost is the entry count.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E = n_r n_c$$

2. Step 2 — Rearrange symbolically for E:

   $$E = n_r n_c$$

3. Step 3 — List the givens: rows (elements of the first set) (n_r) = 7.0000, columns (elements of the second set) (n_c) = 10.0000.

4. Step 4 — Substitute the given values:

   $$E = n_r n_c$$

5. Step 5 — Evaluate:

   $$E = 70.0000\ \text{entries}$$

6. Step 6 — Check: returning E = 70.0000 entries to

   $$E = n_r n_c$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 140.0 — kept a factor of two that cancels in the correct rearrangement.
• 35.0000 — dropped that same factor in the other direction.
• 77.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Discrete Mathematics (Matrix of Relation)

a finite state machine transition table stored as a Boolean matrix Given matrix entries (E) = 381.0 entries; columns (elements of the second set) (n_c) = 9.0000, determine the rows (elements of the first set) (n_r).

Start with the thinking

• The governing relation printed in this handbook section is Entries in a relation (adjacency) matrix.
• Everything except n_r is given, so isolate n_r symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A relation between two finite sets is stored as a Boolean matrix of relation, so the storage cost is the entry count.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E = n_r n_c$$

2. Step 2 — Rearrange symbolically for n_r:

   $$n_{r} = \dfrac{E}{n_c}$$

3. Step 3

   $$List the givens: matrix entries (E) = 381.0 entries, columns (elements of the second set) (n_c) = 9.0000$$

4. Step 4 — Substitute the given values:

   $$n_{r} = \dfrac{381.0}{n_c}$$

5. Step 5 — Evaluate:

   $$n_{r} = 42.3333$$

6. Step 6 — Check: returning n_r = 42.3333 to

   $$E = n_r n_c$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 84.6667 — kept a factor of two that cancels in the correct rearrangement.
• 21.1667 — dropped that same factor in the other direction.
• 46.5667 — rounded an intermediate value before the final step.

Reference: FE Handbook — Discrete Mathematics (Matrix of Relation)

a relation matrix linking pipe nodes to demand nodes Given matrix entries (E) = 180.0 entries; rows (elements of the first set) (n_r) = 13.0000, determine the columns (elements of the second set) (n_c).

Start with the thinking

• The governing relation printed in this handbook section is Entries in a relation (adjacency) matrix.
• Everything except n_c is given, so isolate n_c symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A relation between two finite sets is stored as a Boolean matrix of relation, so the storage cost is the entry count.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E = n_r n_c$$

2. Step 2 — Rearrange symbolically for n_c:

   $$n_{c} = \dfrac{E}{n_r}$$

3. Step 3

   $$List the givens: matrix entries (E) = 180.0 entries, rows (elements of the first set) (n_r) = 13.0000$$

4. Step 4 — Substitute the given values:

   $$n_{c} = \dfrac{180.0}{n_r}$$

5. Step 5 — Evaluate:

   $$n_{c} = 13.8462$$

6. Step 6 — Check: returning n_c = 13.8462 to

   $$E = n_r n_c$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 27.6923 — kept a factor of two that cancels in the correct rearrangement.
• 6.9231 — dropped that same factor in the other direction.
• 15.2308 — rounded an intermediate value before the final step.

Reference: FE Handbook — Discrete Mathematics (Matrix of Relation)

a relation matrix linking survey stations to control points Given rows (elements of the first set) (n_r) = 19.0000; columns (elements of the second set) (n_c) = 8.0000, determine the matrix entries (E) in entries.

Start with the thinking

• The governing relation printed in this handbook section is Entries in a relation (adjacency) matrix.
• Everything except E is given, so isolate E symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A relation between two finite sets is stored as a Boolean matrix of relation, so the storage cost is the entry count.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E = n_r n_c$$

2. Step 2 — Rearrange symbolically for E:

   $$E = n_r n_c$$

3. Step 3 — List the givens: rows (elements of the first set) (n_r) = 19.0000, columns (elements of the second set) (n_c) = 8.0000.

4. Step 4 — Substitute the given values:

   $$E = n_r n_c$$

5. Step 5 — Evaluate:

   $$E = 152.0\ \text{entries}$$

6. Step 6 — Check: returning E = 152.0 entries to

   $$E = n_r n_c$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 304.0 — kept a factor of two that cancels in the correct rearrangement.
• 76.0000 — dropped that same factor in the other direction.
• 167.2 — rounded an intermediate value before the final step.

Reference: FE Handbook — Discrete Mathematics (Matrix of Relation)

a finite state machine transition table stored as a Boolean matrix Given matrix entries (E) = 113.0 entries; columns (elements of the second set) (n_c) = 14.0000, determine the rows (elements of the first set) (n_r).

Start with the thinking

• The governing relation printed in this handbook section is Entries in a relation (adjacency) matrix.
• Everything except n_r is given, so isolate n_r symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A relation between two finite sets is stored as a Boolean matrix of relation, so the storage cost is the entry count.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E = n_r n_c$$

2. Step 2 — Rearrange symbolically for n_r:

   $$n_{r} = \dfrac{E}{n_c}$$

3. Step 3

   $$List the givens: matrix entries (E) = 113.0 entries, columns (elements of the second set) (n_c) = 14.0000$$

4. Step 4 — Substitute the given values:

   $$n_{r} = \dfrac{113.0}{n_c}$$

5. Step 5 — Evaluate:

   $$n_{r} = 8.0714$$

6. Step 6 — Check: returning n_r = 8.0714 to

   $$E = n_r n_c$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 16.1429 — kept a factor of two that cancels in the correct rearrangement.
• 4.0357 — dropped that same factor in the other direction.
• 8.8786 — rounded an intermediate value before the final step.

Reference: FE Handbook — Discrete Mathematics (Matrix of Relation)

a relation matrix linking pipe nodes to demand nodes Given matrix entries (E) = 139.0 entries; rows (elements of the first set) (n_r) = 17.0000, determine the columns (elements of the second set) (n_c).

Start with the thinking

• The governing relation printed in this handbook section is Entries in a relation (adjacency) matrix.
• Everything except n_c is given, so isolate n_c symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A relation between two finite sets is stored as a Boolean matrix of relation, so the storage cost is the entry count.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E = n_r n_c$$

2. Step 2 — Rearrange symbolically for n_c:

   $$n_{c} = \dfrac{E}{n_r}$$

3. Step 3

   $$List the givens: matrix entries (E) = 139.0 entries, rows (elements of the first set) (n_r) = 17.0000$$

4. Step 4 — Substitute the given values:

   $$n_{c} = \dfrac{139.0}{n_r}$$

5. Step 5 — Evaluate:

   $$n_{c} = 8.1765$$

6. Step 6 — Check: returning n_c = 8.1765 to

   $$E = n_r n_c$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 16.3529 — kept a factor of two that cancels in the correct rearrangement.
• 4.0882 — dropped that same factor in the other direction.
• 8.9941 — rounded an intermediate value before the final step.

Reference: FE Handbook — Discrete Mathematics (Matrix of Relation)

a relation matrix linking survey stations to control points Given rows (elements of the first set) (n_r) = 13.0000; columns (elements of the second set) (n_c) = 20.0000, determine the matrix entries (E) in entries.

Start with the thinking

• The governing relation printed in this handbook section is Entries in a relation (adjacency) matrix.
• Everything except E is given, so isolate E symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A relation between two finite sets is stored as a Boolean matrix of relation, so the storage cost is the entry count.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E = n_r n_c$$

2. Step 2 — Rearrange symbolically for E:

   $$E = n_r n_c$$

3. Step 3 — List the givens: rows (elements of the first set) (n_r) = 13.0000, columns (elements of the second set) (n_c) = 20.0000.

4. Step 4 — Substitute the given values:

   $$E = n_r n_c$$

5. Step 5 — Evaluate:

   $$E = 260.0\ \text{entries}$$

6. Step 6 — Check: returning E = 260.0 entries to

   $$E = n_r n_c$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 520.0 — kept a factor of two that cancels in the correct rearrangement.
• 130.0 — dropped that same factor in the other direction.
• 286.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Discrete Mathematics (Matrix of Relation)

a finite state machine transition table stored as a Boolean matrix Given matrix entries (E) = 136.0 entries; columns (elements of the second set) (n_c) = 13.0000, determine the rows (elements of the first set) (n_r).

Start with the thinking

• The governing relation printed in this handbook section is Entries in a relation (adjacency) matrix.
• Everything except n_r is given, so isolate n_r symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A relation between two finite sets is stored as a Boolean matrix of relation, so the storage cost is the entry count.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E = n_r n_c$$

2. Step 2 — Rearrange symbolically for n_r:

   $$n_{r} = \dfrac{E}{n_c}$$

3. Step 3

   $$List the givens: matrix entries (E) = 136.0 entries, columns (elements of the second set) (n_c) = 13.0000$$

4. Step 4 — Substitute the given values:

   $$n_{r} = \dfrac{136.0}{n_c}$$

5. Step 5 — Evaluate:

   $$n_{r} = 10.4615$$

6. Step 6 — Check: returning n_r = 10.4615 to

   $$E = n_r n_c$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 20.9231 — kept a factor of two that cancels in the correct rearrangement.
• 5.2308 — dropped that same factor in the other direction.
• 11.5077 — rounded an intermediate value before the final step.

Reference: FE Handbook — Discrete Mathematics (Matrix of Relation)

a relation matrix linking pipe nodes to demand nodes Given matrix entries (E) = 34.0000 entries; rows (elements of the first set) (n_r) = 6.0000, determine the columns (elements of the second set) (n_c).

Start with the thinking

• The governing relation printed in this handbook section is Entries in a relation (adjacency) matrix.
• Everything except n_c is given, so isolate n_c symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A relation between two finite sets is stored as a Boolean matrix of relation, so the storage cost is the entry count.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E = n_r n_c$$

2. Step 2 — Rearrange symbolically for n_c:

   $$n_{c} = \dfrac{E}{n_r}$$

3. Step 3

   $$List the givens: matrix entries (E) = 34.0000 entries, rows (elements of the first set) (n_r) = 6.0000$$

4. Step 4 — Substitute the given values:

   $$n_{c} = \dfrac{34.0000}{n_r}$$

5. Step 5 — Evaluate:

   $$n_{c} = 5.6667$$

6. Step 6 — Check: returning n_c = 5.6667 to

   $$E = n_r n_c$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 11.3333 — kept a factor of two that cancels in the correct rearrangement.
• 2.8333 — dropped that same factor in the other direction.
• 6.2333 — rounded an intermediate value before the final step.

Reference: FE Handbook — Discrete Mathematics (Matrix of Relation)

a relation matrix linking survey stations to control points Given rows (elements of the first set) (n_r) = 6.0000; columns (elements of the second set) (n_c) = 15.0000, determine the matrix entries (E) in entries.

Start with the thinking

• The governing relation printed in this handbook section is Entries in a relation (adjacency) matrix.
• Everything except E is given, so isolate E symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A relation between two finite sets is stored as a Boolean matrix of relation, so the storage cost is the entry count.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E = n_r n_c$$

2. Step 2 — Rearrange symbolically for E:

   $$E = n_r n_c$$

3. Step 3 — List the givens: rows (elements of the first set) (n_r) = 6.0000, columns (elements of the second set) (n_c) = 15.0000.

4. Step 4 — Substitute the given values:

   $$E = n_r n_c$$

5. Step 5 — Evaluate:

   $$E = 90.0000\ \text{entries}$$

6. Step 6 — Check: returning E = 90.0000 entries to

   $$E = n_r n_c$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 180.0 — kept a factor of two that cancels in the correct rearrangement.
• 45.0000 — dropped that same factor in the other direction.
• 99.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Discrete Mathematics (Matrix of Relation)