# What Are the Properties of the Maps Defined in the Content?

• MHB
• mathmari
In summary, the following holds: -For $a,b\in \mathbb{R}$, it holds that $\text{cost}_a+\text{cost}_b=\text{cost}_{a+b}$.-For $a\in \mathbb{R}$, it holds that $\lambda\text{cost}_a=\text{cost}_{\lambda a}$. -($-(f+g)=(-f)+(-g)$. -$f+f=2f$. -$f+(-f)=\text{cost}_0$. -For $f:\mathbb{R}\right mathmari Gold Member MHB Hey! Let$a\in \mathbb{R}$. We define the map$\text{cost}_a:\mathbb{R}\rightarrow \mathbb{R}$,$x\mapsto a$. We define also$-f:=(-1)f$for a map$f:\mathbb{R}\rightarrow \mathbb{R}$. Let$f:\mathbb{R}\rightarrow\mathbb{R}$be a map and$\lambda\in \mathbb{R}$. Show that: 1. for$a,b\in \mathbb{R}$it holds that$\text{cost}_a+\text{cost}_b=\text{cost}_{a+b}$. 2. for$a\in \mathbb{R}$it holds that$\lambda\text{cost}_a=\text{cost}_{\lambda a}$. 3.$-(f+g)=(-f)+(-g)$. 4.$f+f=2f$. 5.$f+(-f)=\text{cost}_0\$.

Could you give me a hint how we could these? Aren't all of these trivial? (Wondering)

Hey mathmari!

I guess the first one can be shown as follows:
$$\DeclareMathOperator{\cost}{cost} \forall a,b\in\mathbb R,\,\forall x\in\mathbb R : (\cost_a+\cost_b)(x)=\cost_a(x)+\cost_b(x)=a+b=\cost_{a+b}(x)$$
Therefore:
$$\forall a,b\in\mathbb R : \cost_a+\cost_b=\cost_{a+b}$$
(Thinking)

And yes, it does look rather trivial. (Tauri)

Klaas van Aarsen said:
I guess the first one can be shown as follows:
$$\DeclareMathOperator{\cost}{cost} \forall a,b\in\mathbb R,\,\forall x\in\mathbb R : (\cost_a+\cost_b)(x)=\cost_a(x)+\cost_b(x)=a+b=\cost_{a+b}(x)$$
Therefore:
$$\forall a,b\in\mathbb R : \cost_a+\cost_b=\cost_{a+b}$$
(Thinking)

Ahh ok! (Malthe)

In the same way we could show also the other ones, or not? (Wondering)

$$2. \ \ \ \DeclareMathOperator{\cost}{cost} \forall a\in\mathbb R,\,\forall x\in\mathbb R : \lambda \cost_a(x)=\lambda a=\cost_{\lambda a}(x)$$
Therefore:
$$\forall a\in\mathbb R : \lambda\cost_a=\cost_{\lambda a}$$

$$3. \ \ \ \forall x\in\mathbb R : -(f+g)(x)=(-1)(f+g)(x)=(-1)(f(x)+g(x))=(-1)f(x)+(-1)g(x)=(-f)(x)+(-g)(x)$$
Therefore:
$$-(f+g)=(-f)+(-g)$$

$$4. \ \ \ \forall x\in\mathbb R : (f+f)(x)=f(x)+f(x)=2f(x)$$
Therefore:
$$f+f=2f$$

$$5. \ \ \ \forall x\in\mathbb R : (f+(-f))(x)=(f+(-1)f)(x)=f(x)+(-1)f(x)=(1-1)f(x)=0\cdot f(x)=0=\text{cost}_0(x)$$
Therefore:
$$f+(-f)=\text{cost}_0$$ Is everything correct? Could we improve something? (Wondering)

Looks all good to me. (Nod)

Klaas van Aarsen said:
Looks all good to me. (Nod)

Great! Thank you! (Yes)

## 1. What are the different types of maps?

There are several types of maps, including political maps, physical maps, topographic maps, and thematic maps. Political maps show boundaries and locations of countries, states, and cities. Physical maps depict landforms, bodies of water, and natural features. Topographic maps show elevation and terrain. Thematic maps focus on a specific theme, such as population density or climate.

## 2. How are maps created?

Maps are created using a variety of methods, including satellite imagery, aerial photography, and ground surveys. The data collected is then processed and analyzed using Geographic Information Systems (GIS) software to create a digital map. Cartographers also use various tools, such as compasses and rulers, to create hand-drawn maps.

## 3. What are the key elements of a map?

The key elements of a map include a title, legend, scale, compass rose, and grid lines. The title describes the subject of the map, while the legend explains the symbols and colors used. The scale indicates the ratio of distance on the map to actual distance on the ground. The compass rose shows the orientation of the map, and grid lines help with navigation and measuring distances.

## 4. How do maps represent the Earth's spherical surface?

Maps use various projections to represent the Earth's spherical surface on a flat surface. The most commonly used projections are the Mercator projection, which preserves shape and direction but distorts size, and the Robinson projection, which balances size and shape but distorts distance and direction. Other projections include the conic projection and the azimuthal projection.

## 5. What are some common map symbols?

Common map symbols include icons for natural features, such as mountains and lakes, and man-made features, such as roads and buildings. Other symbols represent boundaries, transportation routes, and cultural features. The use of colors, patterns, and labels also helps to convey information on a map.

• Calculus
Replies
9
Views
594
• Calculus
Replies
3
Views
437
• Calculus
Replies
6
Views
576
• Calculus
Replies
3
Views
1K
• Calculus
Replies
20
Views
3K
• Calculus
Replies
3
Views
1K
• Calculus
Replies
1
Views
1K
• Topology and Analysis
Replies
4
Views
245
• Calculus
Replies
9
Views
2K
• Calculus and Beyond Homework Help
Replies
0
Views
577