Determine whether the statement is true or false. Use the following sets of numbers.

$N=$ set of natural numbers.

$Z=$ set of integers

$I=$ set of irrational numbers

$Q=$ set of rational numbers

$\mathbb{R}=$ set of real number

$$-1 \in Z$$

Refer to the table. Which lap has the slowest time?

Write always, sometimes, or never to complete a true statement.

A sequence of a rotation and a dilation will $\underline{\text{?}}$ result in an image that is the same size and orientation as the preimage.

Let $\varphi$ : G $\rightarrow$ GL(V) be a representation of G. The centralizer of $\varphi$ is defined to be the set of all linear transformations, A, from V to itself such that

$$A \varphi(g)=\varphi(g) A$$

for all g $\in$ G (i.e., the linear transformations of V which commute with all $\varphi(g)^{\prime}s$). (a) Prove that a linear transformation A from V to V is in the centralizer of $\varphi$ if and only if it is an FG-module homomorphism from V to itself (so the centralizer of $\varphi$ is the same as the $\left.\operatorname{ring} \operatorname{Hom}_{F G}(V, V)\right)$. (b) Show that if z is in the center of G then $\varphi(z)$ is in the centralizer of $\varphi$. (c) Assume $\varphi$ is an irreducible representation (so V is a simple FG-module). Prove that if H is any finite abelian subgroup of GL(V) such that A$\varphi$(g) = $\varphi$(g)A for all A $\in$ H then H is cyclic (in other words, any finite abelian subgroup of the multiplicative group of units in the ring Hom$_{F G}$(V, V) is cyclic). Hom$_{F G}$(V, V) is a division ring, so this reduces to proving that a finite abelian subgroup of the multiplicative group of nonzero elements in a division ring is cyclic. Show that the division subring generated by an abelian subgroup of any division ring is a field and use Proposition 18. (d) Show that if $\varphi$ is a faithful irreducible representation then the center of G is cyclic. (e) Deduce from (d) that if G is abelian and $\varphi$ is any irreducible representation then G/ker $\varphi$ is cyclic.

Find the least squares regression line for the points $\{(1, 1), (2, 0), (5, −5) \}$. Graph the points and the line.

Let X={a, b, c}. Define a function S from $\mathcal{P}$(X) to the set of bit strings of length 3 as follows. Let Y$\subseteq$X. If a$\in$Y, set $s_1=0$; If a$\notin$Y, set $s_1=1$; If b$\in$Y, set $s_2=0$; If b$\notin$Y, set $s_2=1$; If c$\in$Y, set $s_3=0$; If c$\in$Y, set $s_3=1$. Define S(Y)=$s_1s_2s_3$.

Prove that S is onto.

If H is a subgroup of G and A is an abelian group let $M_{G / H}(A)$ denote the abelian group of all maps from the left cosets gH of H in G to A. (a) Prove that

$$M_{1}^{G}(A) \cong M_{1}^{H}\left(M_{G / H}(A)\right)$$

as H-modules. [If $\left\{g_{i}\right\}_{i \in \mathcal{I}}$ is a choice of left coset representatives of H in G define the correspondence between $f \in M_{1}^{G}(A)$ and

$$F: H \rightarrow M_{G / H}(A)$$

by

$$F(h)\left(g_{i} H\right)=f\left(g_{i} h\right),$$

and check that this is an isomorphism of H-modules. (b) A G-module A such that $H^{n}(H, A)=0$ for all n $\geq$ 1 and all subgroups H of G is called cohomologically trivial. Prove that $M_{1}^{G}(A)$ is a cohomologically trivial for any abelian group A. (c) If G is finite, prove that $\mathbb{Z} G \otimes_{\mathbb{Z}} A$ is cohomologically trivial for all abelian groups A.

the distance east or west from the prime meridian

Find the complement of the following set.

The set {x|x is an integer strictly between 0 and 8} if U is the set of all integers.

Find the distance between the points. $(5,2),(-1,4)$

Make sense of problems. A sale on dog food that costs $10 a bag says buy 1, get one$$$\frac { 1 } { 2 }$ off. What is the true percent discount on the dog food?

The Tethered Satellite discussed in this module is producing 5.00 kV, and a current of 10.0 A flows. (a) What magnetic drag force does this produce if the system is moving at 7.80 km/s? (b) How much kinetic energy is removed from the system in 1.00 h, neglecting any change in altitude or velocity during that time? (c) What is the change in velocity if the mass of the system is 100,000 kg? (d) Discuss the long term consequences (say, a week-long mission) on the space shuttle's orbit, noting what effect a decrease in velocity has and assessing the magnitude of the effect.

Set U: integers from 1 to 20, Set P: prime numbers, Set T: multiples of 3, Set F: multiples of 5. Create a Venn diagram of these sets.

In previous question, let the height h equal $6.00$ m and the speed v$_{boy}$ equal $2.00$ m/s. Assume that the food pack starts from rest.

(a) Tabulate and graph the speed–time graph.

(b) Tabulate and graph the acceleration–time graph. Let the range of time be from $0$ s to $5.00$ s and the time intervals be $0.500$ s.