XOR
As there are a lot of important applications for the exclusiveor (XOR) operation it is dedicated an own blog post. Two of the most important are:

XOR operations are often used in a Linear Feedback Shift Registers (LFSR). LFSR are a crucial step in scrambler / descrambler respectively, which are used in transmitters / receivers

Part of an encryption algorithmus

Last but not least an important part of a halfadder.
The exclusiveor operation xor which is also termed as antivalence is denoted as A ⊕ B = Q. Its truth table is shown below
A 
B 
Q 
0 
0 
0 
0 
1 
1 
1 
0 
1 
1 
1 
0 
CMOS XOR 

Some Applications
Linear Feedback Shift Register (LSFR)
As described above XOR gates are used in Linear Feedback Shift Registers (LFSR). LFSR are often used for test pattern generation. The author selected a simple example of a 3bit LFSR found in a white paper from Texas Instruments.
For this example we need to take a big leap and enter the area of sequential circuits. These are circuits which make use of feedback. The chosen example generates pseudorandom test patterns according to the table below. The registers are feed with the seed value 111, after 8 clock cycles the patterns repeat again. Of course real test pattern generators have a much higher bit width.
CLK 
FF1 
FF2 
FF3 
0 
1 
1 
1 
1 
0 
1 
1 
2 
0 
0 
1 
3 
1 
0 
0 
4 
1 
0 
0 
5 
0 
1 
0 
6 
1 
0 
0 
7 
1 
1 
0 
8 
1 
1 
1 
Halfadder
In the next blogpost we dive deeper into the halfadder application. The halfadder essentialy consists of the two following particular equations for sum (making use of the XOR ) and carry (using a simple AND).
\[ \begin{array}{c} c = x \land y \\ s = x \oplus y \end{array} \]