- #1
Romono
- 5
- 0
Could someone please explain how the image of a set A' ⊆ A is the set: f(A') = {b | b = f(a) for some a ∈ A'}. And how can the complement of A be a subset of A? Forgive my ignorance here, I'm a beginning student of set theory.
Euge said:Hi Romono,
The answer to your first question is by definition. For your second question, the complement of $A$ (in $A$) is the empty set, and the empty set is a subset of $A$.
Euge said:Let's consider an example. Define a function $f : \{1, 2, 3\} \to \{a, b, c\}$ by setting $f(1) = a$, $f(2) = b$, and $f(3) = c$. Since $1$ and $2$ are the only elements of $\{1, 2\}$, $f(\{1,2\}) = \{f(1), f(2)\} = \{a, b\}$.
Romono said:Just to be clear in this example, {a,b} would then be the image, wouldn't it? I think I'm understanding it now...
Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It provides a framework for understanding and analyzing mathematical concepts and structures.
A subset is a set that contains elements that are all part of another set. In other words, all the elements in a subset are also elements of the larger set.
A' ⊆ A means that the complement of set A, denoted by A', is a subset of set A. This means that all the elements in A' are also in A.
The complement of a set is the set of all elements that are not in the original set. In other words, it is the set of all elements in the universal set that are not in the original set.
This is because the complement of a set contains all the elements that are not in the original set. Therefore, the complement of a set can only be a subset of the original set, as the original set contains all the elements in the universe.