answer:Now that we've turned onto Elm Street, you'll notice a stunning example of neoclassical architecture on your left – the City Hall building. Completed in 1925, this impressive structure features a grand staircase, marble columns, and a beautifully restored copper dome. Take a glance up at the dome, and you'll see the city's motto, Unity and Progress, inscribed in bold letters. As we continue down Elm, we're approaching the city's vibrant arts district. You'll see a variety of galleries, studios, and street art installations showcasing the work of local artists. Keep an eye out for the iconic mural on the side of the building just ahead – it's a popular spot for photos. We're making our way toward the city's waterfront area now, where you'll have a chance to take in some breathtaking views of the harbor.

answer:The waterfront area we're approaching is renowned for its fresh seafood. You'll find a variety of excellent seafood restaurants, many of which have been family-owned and operated for generations. Be sure to take note of the historic fish market on your right, where local fishermen bring in their daily catch. If you're looking for a great place to try some of the city's famous seafood, I highly recommend trying one of the eateries in this area. We'll be taking a short break here soon, so you'll have a chance to stretch your legs and sample some of the local cuisine if you'd like. Now, look out at the harbor – you can see the city's iconic lighthouse in the distance. We'll be getting a closer view of it as we continue along the waterfront. The lighthouse has been guiding mariners safely into the harbor since 1850 and is one of the city's most beloved landmarks.

answer:To find the probability that X falls within the interval [0.25, 0.75], we can use the definition of a uniform distribution. A uniform distribution over the interval [0, 1] implies that every point in this interval has an equal probability of being chosen. The probability density function (pdf) of a uniform distribution over [a, b] is given by 1 / (b - a). In this case, a = 0 and b = 1, so the pdf is 1 / (1 - 0) = 1. To calculate the probability that X falls within the interval [0.25, 0.75], we need to integrate the pdf over this interval. This can be expressed as the integral of 1 from 0.25 to 0.75. Therefore, the probability is the integral from 0.25 to 0.75 of 1 dx. Evaluating this integral gives (0.75 - 0.25) * 1 = 0.5. This is the probability that X falls within the interval [0.25, 0.75]. Notice that this result makes intuitive sense, since the interval [0.25, 0.75] covers half of the total interval [0, 1], and the distribution is uniform, meaning that every point is equally likely. Thus, it's expected that the probability of falling within this half of the interval would be 0.5.

answer:To show that R is an equivalence relation, we need to verify that it satisfies three properties: reflexivity, symmetry, and transitivity. Reflexivity states that for any real number x, x R x. This is true because x - x = 0, and 0 is an integer. Therefore, x R x holds for any real number x. Symmetry states that if x R y, then y R x. Let's assume x R y, meaning x - y is an integer. Then, y - x = -(x - y), which is also an integer, since the negative of an integer is also an integer. Therefore, y R x holds whenever x R y, establishing symmetry. Transitivity states that if x R y and y R z, then x R z. Assume x R y and y R z. Then, x - y and y - z are both integers. Adding these two expressions, we get x - z = (x - y) + (y - z), which is also an integer since the sum of two integers is an integer. Therefore, x R z holds whenever x R y and y R z, establishing transitivity. Since R satisfies all three properties - reflexivity, symmetry, and transitivity - it is an equivalence relation. This equivalence relation partitions the set of real numbers into equivalence classes, where two real numbers belong to the same class if their difference is an integer. For example, all integers belong to one class, all real numbers of the form n + 0.5, where n is an integer, belong to another class, and so on.