Demonstrating set properties of a map

[Baris Kalfa]

Hello, could you please help me regarding this question about a certain map (application).

Homework Statement

Demonstrate if A₁ ⊂ A₂ →ƒ(A₁) ⊂ ƒ(A₂)

2. Homework Equations
ƒ:A→B is a map
A₁, A₂⊂ A

The Attempt at a Solution

first assumed that (A₁∪A₂)⊆A
⇒ (ƒ(A₁) ∪ ƒ(A₂))⊆ ƒ(A)
then if A₁ ⊂ A₂
∴ ƒ(A₁) ⊂ ƒ(A₁)

I don't know if this demonstration is satisfying enough. I'm missing something related to properties of a function.

 

[Baris Kalfa]

I have also tried it this way,
I have assumed that
ƒ(A₁⊂ A₂) = ƒ(A₁) ⊂ ƒ(A₂)
∴A₁⊂ A₂→ƒ(A₁⊂A₂)
Thus, A₁⊂ A₂→ƒ(A₁)⊂ƒ(A₂)

This one looks better, doesn't it?

 

[Ray Vickson]

Hello, could you please help me regarding this question about a certain map (application).

Homework Statement

Demonstrate if A₁ ⊂ A₂ →ƒ(A₁) ⊂ ƒ(A₂)

2. Homework Equations
ƒ:A→B is a map
A₁, A₂⊂ A

The Attempt at a Solution

first assumed that (A₁∪A₂)⊆A
⇒ (ƒ(A₁) ∪ ƒ(A₂))⊆ ƒ(A)
then if A₁ ⊂ A₂
∴ ƒ(A₁) ⊂ ƒ(A₁)

I don't know if this demonstration is satisfying enough. I'm missing something related to properties of a function.

None of you "proofs" is anything of the kind (unless you are citing some results in your textbook or course notes that have already established the relationships you are using). As far as I can see you have just written down some relationships without any verification whatsoever.

Go back to the basics: (i) for a set $A$ and a function $f$, what is the definition of $f(A)?$ (ii) for two sets $A_1, A_2$, what is meant by the assertions $A_1 \subset A_2$ or $f(A_1) \subset f(A_2)?$ You need to look at individual elements $f(x)$ for $x \in A.$