# Proof involving functions and intersection

• PCSL
In summary, the statement "f(\cap T_\alpha)=\cap f(T_\alpha) for all choices of (T_\alpha) \alpha \in \lambda" is only true if f is one-to-one. To prove this, assume f is one-to-one and show that y is an element of f(\cap T_\alpha) if and only if it is an element of \cap f(T_\alpha). For the opposite direction, show that if f is not one-to-one, there exists a counterexample. Additional resources for reviewing functions, order isomorphisms, etc. may be helpful, as the book being used may not provide enough examples.
PCSL
Prove that $$f(\cap T_\alpha)=\cap f(T_\alpha)$$ for all choices of $$(T_\alpha) \alpha \in \lambda$$ if and only if f is one-to-one.

I've been working on this on and off for a day and have nothing to show for it... Any help pointing me in the right direction would be appreciated.

Also, more important then this question, can any of you recommend additional resources to review this type of stuff, mostly functions, order isomorphisms, etc. The book I am using does not give as many examples as I'd like.

It's not really that hard. First assume f is one-to-one. Now can you show that if y is an element of $f(\cap T_\alpha)$ then it is an element of $\cap f(T_\alpha)$ and vice versa? For the opposite direction I'd do the contrapositive. Show if f is not one-to-one then you can find a counterexample.

## 1. What is a function in mathematics?

A function is a mathematical relationship between two sets of values, where each input has a unique output. It can be represented as a set of ordered pairs, a graph, or an equation.

## 2. How do you prove that two functions intersect?

To prove that two functions intersect, you must find the values of the variables that make both functions equal. This can be done by setting the two functions equal to each other and solving for the variables.

## 3. Can two functions have more than one point of intersection?

Yes, two functions can have more than one point of intersection. This occurs when there are multiple values of the variables that make both functions equal.

## 4. What is the significance of proving functions intersect?

Proving that functions intersect can provide important information about the relationships between the two functions. It can also be useful in solving equations and finding solutions to real-world problems.

## 5. Are there any special cases when proving functions intersect?

Yes, there are special cases when proving functions intersect. One example is when dealing with vertical lines, as they will intersect with any other function at exactly one point. Additionally, functions with the same slope will have infinite points of intersection.

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