Explain how the symbols for subset and proper subset are related to the symbols < and for ≤ numbers.
provide an example

The line under the symbols will have the same effect for the subset and proper subset symbols and the less than and less than or equal symbols.

By concept, a proper subset is the set that will have some but not all of the values of a given set, for example:

Imagine we had the sets:

A={a,e,i,o,u}

B={a,o,u}

C={a,e,i,ou}

we can say that

B⊂A   (B is a proper subset of A)

but we cannot say that:

C⊂A

because B has some elements of A, while C has all the elements in A, so C is not a proper subset of A.

Now, a subset can contain some or all of the elements contained in another set.

we can for sure say that:

B⊆A   and also that C⊆A

Because C has all the elements of A, so it fits into the subset definition.

Comparing this to the < and ≤ symbols, the < symbol means that a value will be less than another value. This doesn't include the greater value, for

example, we can say that:

2<5

but we cannot say that 5<5 because they are both the same. That statement is false.

On the other hand, the ≤ stands for, less than or equal to. This symbol can be used when a number is less than another one or equal to it, for example, we can say that:

2≤5 and we can also say that 5≤5 because they are the same and the symbol does include the actual value of 5.

So as you may see, the relation is that the line under the symbol includes the values or sets while if the symbols don't have a line under them, this means that the greater value or the original set is not to be included.

the transverse axis of a hyperbola is the line that contains the two vertices and the two focuses.

in this example (hyperbola of equation x^2/2 - y^2/4 = 1), the transverse axis is the  x-axis  . 

step-by-step explanation: