a) Show that congruence modulo m is an equivalence relation whenever m is a positive integer. b) Show that the relation {(a, b) | a ≡ ±b (mod 7)} is an equivalence relation on the set of integers.

Show that the argument form with premises p1, p2, . . . , pn and conclusion q → r is valid if the argument form with premises p1, p2, . . . , pn, q, and conclusion r is valid.

Let R be an equivalence relation on a set A. Prove each of the statements directly from the definitions of equivalence relation and equivalence class. For all a, b and c in A, if b R c and

$$c \in [ a ]$$

then

$$b \in [ a ]$$

a. Let R be the relation of congruence modulo 3. Which of the following equivalence classes are equal?

$$[ 7 ] , [ - 4 ] , [ - 6 ] , [ 17 ] , [ 4 ] , [ 27 ] , [ 19 ]$$

b. Let R be the relation of congruence modulo 7. Which of the following equivalence classes are equal?

$$[ 35 ] , [ 3 ] , [ - 7 ] , [ 12 ] , [ 0 ] , [ - 2 ] , [ 17 ]$$

Consider a relation r(A, B, C), with an index on attribute A. Give an example of a query that can be answered by using the index only, without looking at the tuples in the relation. (Query plans that use only the index, without accessing the actual relation, are called index-only plans.)

Show that any nonempty set of integers that is closed under subtraction must also be closed under addition.

A projectile is launched over level ground at an initial elevation angle of $\theta$. An observer standing at the launch site sees the projectile at the point of its highest elevation and measures the angle $\phi$ shown in Figure 3-38. Show that $\tan \phi=\frac{1}{2} \tan \theta$. (Ignore any effects due to air resistance.)

During which stage of the cell cycle is DNA duplicated?

a. G1$\\$ b. G2$

$$c. M$$

$ d. S

Prove that the relation of congruence is an equivalence relation.

Uno de los puntos de regulación del ciclo celular opera en el pasaje de:

a) G1 a G2

b) G1 a M

c) G1 a S

d) G2 a G1

Which of the following is the correct order for the phases of the cell cycle?

a. S, G2, M, and G1

b. G2, S, G1, and M

c. M, S, G1, and G2

d. G1, S, G2, and M

Prove the following statement:

If

$$A=\left[\begin{array}{ccc} a&b\\ c&d \end{array}\right]$$

and $ad-bc\neq0$, then

$$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{ccc} d&-b\\ -c&a \end{array}\right].$$

Prove that the inverse of the matrix

$$A=\begin{bmatrix}a & b\\c & d\end{bmatrix}$$

is

$$A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}$$

provided ad

$$a d - b e \neq 0.$$

Which represents the correct sequence of stages in the cell cycle?

A. G1, G2, S, M

B. G1, G2,M, S

C. G1, M, G2, S

D. G1, S, G2, M