Draw all nonisomorphic simple graphs having three vertices.

If $\alpha$ is a bit string, let $C(\alpha)$ be the maximum number of consecutive 0’s in $\alpha$. [Examples: $C(10010) = 2, C(00110001) = 3$.] Let $S_n$ be the number of n-bit strings $\alpha$ with $C(\alpha) \leq 2$. Develop a recurrence relation for $S_1, S_2, . . . .$

The following problem, two matrices A and B and two numbers c and d are given. Compute the matrix cA + dB.

$$\mathbf{A}=\left[\begin{array}{rrr} 2 & 0 & -3 \\ -1 & 5 & 6 \end{array}\right], \mathbf{B}=\left[\begin{array}{rrr} -2 & 3 & 1 \\ 7 & 1 & 5 \end{array}\right], c=5, d=-3$$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $\lim _{x \rightarrow 2}\left(\frac{x}{x+1}+\frac{3}{x-1}\right)=\lim _{x \rightarrow 2} \frac{x}{x+1}+\lim _{x \rightarrow 2} \frac{3}{x-1}$

Solve the equation of quadratic form.

$$x^{1 / 3}-x^{1 / 6}-6=0$$

$C_{0}, C_{1}, C_{2}, \ldots$ denotes the sequence of Catalan numbers. Prove that $C_{n} \geq \frac{4^{n-1}}{n^{2}}$ for all $n \geq 1$.

When the saddle point exists, find the strategies producing it and the value of the game. Identify any games that are strictly determined.

$$\left[\begin{array}{rr} 8 & 6 \\ 12 & -4 \end{array}\right]$$

Your friend asks you to explain how the product rule for independent events differs from the product rule for dependent events. How would you respond?

For each of the systems, find the equilibrium points and analyze their stability.

$$\begin{array}{l} x^{\prime}=-x+3 z \\ y^{\prime}=-y+2 z \\ z^{\prime}=x^{2}-2 z \end{array}$$

Solve the equation by using the Square Root Property. (m - 12)$^{2}$ = 400

The following problem illustrate ways in which the algebra of matrices is not analogous to the algebra of real numbers. Find a 2 x 2 matrix A with each element +1 or -1 such that $\mathbf{A}^{2}=\mathbf{0}$. The formula of earlier problem may be helpful.

Write the number in i-form. $\sqrt{-48}$

Solve the equation of quadratic form. $x^{4}-10 x^{2}+25=0$

Solve each equation. $\frac{x}{3}-7=6-\frac{3 x}{4}$