Prove that G or bar G must be nonplanar if G has 11 vertices. The exact problem is the following: "Let G be a graph with 11 vertices. Prove that G or bar G must be nonplanar. I went to my professor to discuss the proof, but she said it didn't quite work, that something was "off" about it. The following is the proof I presented: Let G be a graph with 11 vertices. It is clear that |E(G)|+|E(bar G)|=|E(K11)|=55. Assume, for the sake of contradiction, that both G and bar G are nonplanar. Then |E(G)|+|E(bar G)|<=(3|V(G)|−6)+(3|V(bar G)|−6)=54.Rightarrow Leftarrow Therefore, either G or bar G is nonplanar. So, where exactly does this proof fail? Any tips on the aesthetics of the proof would also be greatly appreciated.

What is the range of the function ${\displaystyle y=x^2}$?

The domain and range of $y=\sin <!--\; \; -->x$ is
A)$R,[0,\mathrm{\infty <!--\; \infty \; -->}]$
B)$R,[-<!--\; -\; -->1,1]$
C)$R,[-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},\mathrm{\infty <!--\; \infty \; -->}]$
D)$R,[1,\mathrm{\infty <!--\; \infty \; -->}]$

Mean of the squares of the deviations from mean is called the:
Variance
Standard deviation
Quartile deviation
Mode

How to evaluate P(10,2)?

What is the effect of wind speed on evaporation?

What is the range of a linear function?

Find the range of ${\displaystyle \frac{x^2}{1-x^2}}$

State the main limitations of statistics.

What is the range of the function y = x?

Tell whether the statement is true or false : A data always has a mode.

In most cases,_____variables are not considered to be equal unless they are exactly the same.

A local newspaper in a large city wants to assess support for the construction of a highway bypass around the central business district to reduce downtown traffic. They survey a random sample of 1152 residents and find that 543 of them support the bypass. Construct and interpret a 95% confidence interval to estimate the proportion of residents who support construction of the bypass.

$3+3x3-3+3=?$What is the right answer and how?

Define lateral displacement.