# Verify that the Euler's formula is correct?

Proof by induction below:

(it is long, but not very complicated)

It is necessary to define the environment before we can prove Euler's Formula for Planar Graphs. For the purposes of this discussion, a Planar Graph is:

consists of three different kinds of elements:

Additionally

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Considering the aforementioned:

One vertex, one edge (with both ends connected to the vertex), and two faces—one inside the loop the edge creates when it loops back to the vertex and the other outside—are the components of a minimal planar graph.

Note that Option 3—Adding a Face Without Adding a Vertex or Edge—is not available.

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(see below)

I have taken that you really did mean "verify" which implies that there should have been a sample to verify. It is also possible that you really meant "prove"; if so re-submit this question with proper terminology.

I have also assumed that, since this was asked under the Geometry topic, you meant Euler's Formula relating Vertices, Edges, and Faces for planar figures in Euclidean space (Euler has several formulae in different areas).

Here is a sample for verification:

Euler's Formula says

and for the sample (above) since

we have verified Euler's Formula (for this example).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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