a relation matrix linking survey stations to control points Given rows (elements of the first set) (n_r) = 9.0000; columns (elements of the second set) (n_c) = 17.0000, 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) = 9.0000, columns (elements of the second set) (n_c) = 17.0000.

4. Step 4 — Substitute the given values:

   $$E = n_r n_c$$

5. Step 5 — Evaluate:

   $$E = 153.0\ \text{entries}$$

6. Step 6 — Check: returning E = 153.0 entries to

   $$E = n_r n_c$$

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

Why the other options are there

• 306.0 — kept a factor of two that cancels in the correct rearrangement.
• 76.5000 — dropped that same factor in the other direction.
• 168.3 — 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) = 232.0 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

   $$List the givens: matrix entries (E) = 232.0 entries, columns (elements of the second set) (n_c) = 9.0000$$

4. Step 4 — Substitute the given values:

   $$n_{r} = \dfrac{232.0}{n_c}$$

5. Step 5 — Evaluate:

   $$n_{r} = 25.7778$$

6. Step 6 — Check: returning n_r = 25.7778 to

   $$E = n_r n_c$$

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

Why the other options are there

• 51.5556 — kept a factor of two that cancels in the correct rearrangement.
• 12.8889 — dropped that same factor in the other direction.
• 28.3556 — 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) = 353.0 entries; rows (elements of the first set) (n_r) = 18.0000, 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

   $$List the givens: matrix entries (E) = 353.0 entries, rows (elements of the first set) (n_r) = 18.0000$$

4. Step 4 — Substitute the given values:

   $$n_{c} = \dfrac{353.0}{n_r}$$

5. Step 5 — Evaluate:

   $$n_{c} = 19.6111$$

6. Step 6 — Check: returning n_c = 19.6111 to

   $$E = n_r n_c$$

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

Why the other options are there

• 39.2222 — kept a factor of two that cancels in the correct rearrangement.
• 9.8056 — dropped that same factor in the other direction.
• 21.5722 — 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) = 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) = 19.0000, 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:

   $$E = 190.0\ \text{entries}$$

6. Step 6 — Check: returning E = 190.0 entries to

   $$E = n_r n_c$$

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

Why the other options are there

• 380.0 — kept a factor of two that cancels in the correct rearrangement.
• 95.0000 — dropped that same factor in the other direction.
• 209.0 — 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) = 113.0 entries; columns (elements of the second set) (n_c) = 5.0000, 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

   $$List the givens: matrix entries (E) = 113.0 entries, columns (elements of the second set) (n_c) = 5.0000$$

4. Step 4 — Substitute the given values:

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

5. Step 5 — Evaluate:

   $$n_{r} = 22.6000$$

6. Step 6 — Check: returning n_r = 22.6000 to

   $$E = n_r n_c$$

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

Why the other options are there

• 45.2000 — kept a factor of two that cancels in the correct rearrangement.
• 11.3000 — dropped that same factor in the other direction.
• 24.8600 — 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) = 346.0 entries; rows (elements of the first set) (n_r) = 16.0000, 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

   $$List the givens: matrix entries (E) = 346.0 entries, rows (elements of the first set) (n_r) = 16.0000$$

4. Step 4 — Substitute the given values:

   $$n_{c} = \dfrac{346.0}{n_r}$$

5. Step 5 — Evaluate:

   $$n_{c} = 21.6250$$

6. Step 6 — Check: returning n_c = 21.6250 to

   $$E = n_r n_c$$

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

Why the other options are there

• 43.2500 — kept a factor of two that cancels in the correct rearrangement.
• 10.8125 — dropped that same factor in the other direction.
• 23.7875 — 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) = 11.0000; columns (elements of the second set) (n_c) = 4.0000, 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) = 11.0000, columns (elements of the second set) (n_c) = 4.0000.

4. Step 4 — Substitute the given values:

   $$E = n_r n_c$$

5. Step 5 — Evaluate:

   $$E = 44.0000\ \text{entries}$$

6. Step 6 — Check: returning E = 44.0000 entries to

   $$E = n_r n_c$$

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

Why the other options are there

• 88.0000 — kept a factor of two that cancels in the correct rearrangement.
• 22.0000 — dropped that same factor in the other direction.
• 48.4000 — 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) = 90.0000 entries; columns (elements of the second set) (n_c) = 17.0000, 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

   $$List the givens: matrix entries (E) = 90.0000 entries, columns (elements of the second set) (n_c) = 17.0000$$

4. Step 4 — Substitute the given values:

   $$n_{r} = \dfrac{90.0000}{n_c}$$

5. Step 5 — Evaluate:

   $$n_{r} = 5.2941$$

6. Step 6 — Check: returning n_r = 5.2941 to

   $$E = n_r n_c$$

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

Why the other options are there

• 10.5882 — kept a factor of two that cancels in the correct rearrangement.
• 2.6471 — dropped that same factor in the other direction.
• 5.8235 — 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) = 335.0 entries; rows (elements of the first set) (n_r) = 10.0000, 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

   $$List the givens: matrix entries (E) = 335.0 entries, rows (elements of the first set) (n_r) = 10.0000$$

4. Step 4 — Substitute the given values:

   $$n_{c} = \dfrac{335.0}{n_r}$$

5. Step 5 — Evaluate:

   $$n_{c} = 33.5000$$

6. Step 6 — Check: returning n_c = 33.5000 to

   $$E = n_r n_c$$

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

Why the other options are there

• 67.0000 — kept a factor of two that cancels in the correct rearrangement.
• 16.7500 — dropped that same factor in the other direction.
• 36.8500 — 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) = 9.0000; columns (elements of the second set) (n_c) = 4.0000, 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) = 9.0000, columns (elements of the second set) (n_c) = 4.0000.

4. Step 4 — Substitute the given values:

   $$E = n_r n_c$$

5. Step 5 — Evaluate:

   $$E = 36.0000\ \text{entries}$$

6. Step 6 — Check: returning E = 36.0000 entries to

   $$E = n_r n_c$$

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

Why the other options are there

• 72.0000 — kept a factor of two that cancels in the correct rearrangement.
• 18.0000 — dropped that same factor in the other direction.
• 39.6000 — rounded an intermediate value before the final step.

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