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The most widely used versions of Photoshop are Photoshop CS and Photoshop CS2. The latest version is Photoshop CS3, with many of its features being made available to Photoshop CS2 users as downloadable updates. In this book, we concentrate on Photoshop CS2 and CS3 (you can download a free trial of Photoshop CS3 if you’re considering buying Photoshop CS3) and cover all the things you need to know to get your images looking their best.

**Figure 1-1:** A series of circles illustrate how layers can be blended into each other to create a graduated background.

Weaving Layers into Your Images

At the heart of Photoshop is the layer. You can add a new layer to an image and use any of Photoshop’s built-in tools to create a new layer, apply effects and filters, cut, copy, and paste, rotate, create shapes, and move objects. In Figure 1-2, you can see a few of the layers that are part of the original image. When you work in Photoshop, you can create as many layers as you want to work with as separate entities. In this example, there are three layers: a background layer, a text layer, and a foreground (a layer containing the main subject matter in the photo).

You can create layers by selecting a layer in the Layers panel, right-clicking on the layer, and choosing the New Layer command. You can apply several types of effects to a layer as well. For example, to create a layer mask, you simply select the layer you want to mask in the Layers panel and then click the Layer Mask icon, as shown in Figure 1-2, right. Selecting a layer mask enables you to mask out areas in the layer and reveal areas that you don’t want to appear in your image. For example, perhaps you’d like to remove a part of the backdrop but retain the subject matter and perhaps have it appear as a textured blur. To create such an effect, you would create a layer mask for the backdrop layer and mask out the background elements you want to retain, and then use the Blur filter to create a mask that is used to mask the layer with those retained elements (see Figure 1-3).

The Blur filter can be used to blur several layers, but there are various other effects you can use to retouch or otherwise manipulate layers. To view the available options, click the small button in the bottom-right corner of the Blur dialog

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Adobe Photoshop CS6 is the most recent version of Photoshop. It was released in April 2015. If you’ve used the software, you might be surprised to learn that many features are incompatible.

Adobe Photoshop 7 is an older version of Photoshop. It was released in June 2004 and was designed to be more intuitive and easy to use for novice users. This version is similar to Photoshop Elements 7, but there are some different features like the ability to import TIFF files.

Adobe Photoshop SE is an older version of Photoshop. It was released in January 2004, and was a stand-alone version of Photoshop. In addition to the standard features, SE includes a lot of tools and filters that you may need to use while creating images. It was discontinued in March 2009.

Adobe Photoshop Lightroom is a software that works well with Photoshop. It combines features of traditional editing programs like Photoshop and features of post-processing programs like Lightroom. It is a cross-platform solution and works well on both Mac and Windows computers.

Adobe Photoshop for Android is available for tablets and smartphones. However, it is not an alternative to Photoshop as it’s an Android app that is meant to be used with mobile devices. It is meant for image processing, not editing.

I work with photographers, graphic designers, web designers, Discord emoji creators and meme-makers all day long. I also give Photoshop CS5+ for photographers, graphic designers, web designers, Discord emoji creators and meme-makers self-paced online training classes. I’ve taught hundreds of students during this time. I can share with you my tips, tricks and techniques so you can become a Photoshop expert as well.

When it comes to using Photoshop for graphic designers and photographers, the three most popular questions I get asked in training classes are:

It has so many features what should I use Photoshop for? Adobe Photoshop is the most powerful software for the industry. It has so many features what should I use Photoshop for?

Adobe Photoshop is the most powerful software for the industry. It has so many features what should I use Photoshop for? I hear it’s very expensive, what can I use instead? I’m sure you heard that Photoshop is very expensive. If you are a working graphic designer, web designer, photographer or an editor, then you should know that Photoshop is the most expensive and powerful software
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Semiconductor devices are commonly found in modern electronic products. Semiconductor devices vary in the number and density of electrical components. Discrete semiconductor devices generally contain one type of electrical component, e.g., light emitting diode (LED), small signal transistor, resistor, capacitor, inductor, and power metal oxide semiconductor field effect transistor (MOSFET). Integrated semiconductor devices typically contain hundreds to millions of electrical components. Examples of integrated semiconductor devices include microcontrollers, microprocessors, charged-coupled devices (CCDs), solar cells, and digital micro-mirror devices (DMDs).
Semiconductor devices perform a wide range of functions such as signal processing, high-speed calculations, transmitting and receiving electromagnetic signals, controlling electronic devices, transforming sunlight to electricity, and creating visual projections for television displays. Semiconductor devices are found in the fields of entertainment, communications, power conversion, networks, computers, and consumer products. Semiconductor devices are also found in military applications, aviation, automotive, industrial controllers, and office equipment.
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Proof of triangle inequality using Jensen’s inequality.

The problem is:

Let $f:[0,\infty] \rightarrow [0,\infty]$ be the function defined as $f(x)=x^{1/2}$, then prove that the following inequality holds.
$$\left|\sum_{k=1}^n a_k\right| \leq \sum_{k=1}^n |a_k|$$
where $a_k \in \mathbb{R}$.

This is a homework problem from Introduction to Mathematical Analysis by Courant & Robbins.
I attempted this problem but got stuck. This is what I have so far:
$$a_k \in \mathbb{R} \implies a_k^2 \leq a_k^2$$
$$\implies \sum_{k=1}^n a_k^2 \leq \sum_{k=1}^n a_k^2$$
I’m not sure what to do next. It looks like this might be Jensen’s inequality, but I don’t know how to prove it that way or if it’s even right. Any help would be greatly appreciated, thanks.

It is not true in general. Take $a_k = k$ for $k \in \mathbb{N}$. Then $\sum_{k=1}^n a_k=n$ but $\left|\sum_{k=1}^n a_k\right| = n+1$.
But you can use a little trick for such problems. From your first observation, it follows that
$$\sum_{k=1}^n a_k \geq \sqrt{\sum_{k=1}^n a_k^2}\tag{*}.$$
Thus
$$\left|\sum_{k=1}^n a_k\right| = \sum_{k=1}^n |a_k| \leq \sum_{k=1}^n a_k$$
as required.

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