Observations
We continue considering the painting problem, but now we also use observations. We can use the assumptions to derive additional information. An observation is a direct consequence of the assumptions, or of preceding observations. For example, we can calculate the area to be painted using the dimensions of the rooms. In this case our observations about the floor area are based on two assumptions; the dimensions of the living room and the dimensions of the kitchen. We can shorten the derivation by writing this down as an observation. Furthermore, we can use the observations several times without the need to justify them every time.
The structure is the same as earlier, with small additions. The observations are listed after the assumptions and denoted by either an id in square brackets ([1], [2], ...; [obs1], [obs2], ...) or the plus sign \(+\).
We look at the same assignment as earlier and make some observation:
Assignment:
Step 1:
Structured derivation:
Earlier when we did not use the observations and we wrote the Pelle's paint problem as follows.
| \(\bullet\) | How much paint is needed to cover the floor when |
| (a) | the floor is painted twice and |
| (b) | the living room is \(5m\) long and \(3m\) wide and |
| (c) | the kitchen is \(2m\) long and \(1m\) wide and |
| (d) | the average paint consumption is \(\frac{1l}{5m^{2}}\) |
| \(\Vdash\) | amount of paint needed |
| \(=\) | {amount of paint needed(\(l\) ) = 2 \(\cdot\) floor area (\(m^{2}\) ) . average paint consumption (\(\frac{l}{m^{2}}\) ), these are written as assumptions (a), (d)} |
| \(2\cdot floor area (m^{2})\cdot\frac{1l}{5m^{2}}\) | |
| \(=\) | {floor area = floor area of kitchen + floor area of living room. The formula for area is length \(\cdot\) width, we get the dimensions from assumptions (b), (c)} |
| \(2\cdot(5m\cdot3m+2m\cdot1m)\cdot\frac{1l}{5m^{2}}\) | |
| \(=\) | {We do the calculations} |
| \(2\cdot17m^{2}\cdot\frac{1l}{5m^{2}}\) | |
| \(\approx\) | {We simplify \(m^{2}\) } |
| \(7l\) | |
| \(\square\) |
If we rather calculate the floor area beforehand, witch is now calculated in the middle of derivation, we can do this by adding following observations:
- The floor area of the living room
- The floor area of the kitchen
- The total floor area (living room + kitchen)
| \(\bullet\) | How much paint is needed to cover the floor when |
| (a) | the floor is painted twice and |
| (b) | the living room is \(5m\) long and \(3m\) wide and |
| (c) | the kitchen is \(2m\) long and \(1m\) wide and |
| (d) | the average paint consumption is \(\frac{1l}{5m^{2}}\) |
| \(\Vdash\) | amount of paint needed |
| \(=\) | {amount of paint needed(\(l\) ) = 2 \(\cdot\) floor area (\(m^{2}\) ) . average paint consumption (\(\frac{l}{m^{2}}\) ), these are written as assumptions (a), (d)} |
| \(2\cdot floor area (m^{2})\cdot\frac{1l}{5m^{2}}\) | |
| \(=\) | {floor area = floor area of kitchen + floor area of living room. The formula for area is length \(\cdot\) width, we get the dimensions from assumptions (b), (c)} |
| \(2\cdot(5m\cdot3m+2m\cdot1m)\cdot\frac{1l}{5m^{2}}\) | |
| \(=\) | {We do the calculations} |
| \(2\cdot17m^{2}\cdot\frac{1l}{5m^{2}}\) | |
| \(\approx\) | {We simplify \(m^{2}\) } |
| \(7l\) | |
| \(\square\) |
After the observations identifier ([1]), the first part of an observation is the justification ...
| \(\bullet\) | How much paint is needed to cover the floors when |
| (a) | the floor is painted twice and |
| (b) | the living room is \(5m\) long and \(3m\) wide and |
| (c) | the kitchen is \(2m\) long and \(1m\) wide and |
| (d) | the average paint consumption is \(\frac{1l}{5m^{2}}\) |
| [1] | {We get the floor area of the living room from assumption (b)} |
... and the second part is the actual observation.
In this example the observations were simple and the calculations were easy. If a longer calculation is needed for the observation, it can be written as a separate derivation.
| \(\bullet\) | How much paint is needed to cover the floors when |
| (a) | the floor is painted twice and |
| (b) | the living room is \(5m\) long and \(3m\) wide and |
| (c) | the kitchen is \(2m\) long and \(1m\) wide and |
| (d) | the average paint consumption is \(\frac{1l}{5m^{2}}\) |
| [1] | {We get the floor area of the living room from assumption (b)} |
| The floor area of the living room is \(5 m \cdot 3 m = 15 m^{2}\) |
Now we add two more observations.
| \(\bullet\) | How much paint is needed to cover the floor when |
| (a) | the floor is painted twice and |
| (b) | the living room is \(5m\) long and \(3m\) wide and |
| (c) | the kitchen is \(2m\) long and \(1m\) wide and |
| (d) | the average paint consumption is \(\frac{1l}{5m^{2}}\) |
| [1] | {We get the floor area of the living room from assumption (b)} |
| The floor area of the living room is\(5 m \cdot 3 m = 15 m^{2}\) | |
| [2] | {We get the floor area of the kitchen from assumption (c)} |
| The floor area of the kitchen is \(2 m \cdot 1 m = 2 m^{2}\) | |
| [3] | {We get the total area to be painted from observations [1] and [2]} |
| The total area to be painted is \(15 m^{2} + 2 m^{2} = 17 m^{2}\) |
When all the observations are written down, we can continue with the actual derivation to solve the problem. When we use infomation in observation, we refer to observations identifier in motivation.
| \(\bullet\) | How much paint is needed to cover the floor when |
| (a) | the floor is painted twice and |
| (b) | the living room is \(5m\) long and \(3m\) wide and |
| (c) | the kitchen is \(2m\) long and \(1m\) wide and |
| (d) | the average paint consumption is \(1l / 5m^{2} \) |
| [1] | {We get the floor area of the living room from assumption (b)} |
| The floor area of the living room is\(5 m \cdot 3 m = 15 m^{2}\) | |
| [2] | {We get the floor area of the kitchen from assumption (c)} |
| The floor area of the kitchen is \(2 m \cdot 1 m = 2 m^{2}\) | |
| [3] | {We get the total area to be painted from observations [1] and [2]} |
| The total area to be painted is \(15 m^{2} + 2 m^{2} = 17 m^{2}\) | |
| \(\Vdash\) | amount of paint needed |
| \(=\) | {amount of paint needed (\(l\) ) = 2 \(\cdot\) floor area (\(m^{2}\) ) \(\cdot\) average paint consumption (\(\frac{l}{m^{2}}\) ), these are written as assumptions (a), (d)} |
| \(2\cdot floor area (m^{2})\cdot\frac{1l}{5m^{2}}\) | |
| \(=\) | {the floor total floor area is written as observation [3]} |
| \(2\cdot17m^{2}\cdot\frac{1l}{5m^{2}}\) | |
| \(\approx\) | {We simplify \(m^{2}\) } |
| \(7l\) | |
| \(\square\) |
