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+====== Bitwise AND (&), Bitwise OR (|), Bitwise XOR (^) ======
+===== Bitwise AND (&) =====
+The bitwise operators perform their calculations at the bit level of variables. They help solve a wide range of common programming problems. Much of the material below is from an excellent tutorial on bitwise math wihch may be found here.
+===== Description and Syntax =====
+Below are descriptions and syntax for all of the operators. Further details may be found in the referenced tutorial.
+===== Bitwise AND (&) =====
+The bitwise AND operator in C++ is a single ampersand, &, used between two other integer expressions. Bitwise AND operates on each bit position of the surrounding expressions independently, according to this rule: if both input bits are 1, the resulting output is 1, otherwise the output is 0. Another way of expressing this is:
+<code cpp>
+  0  1  1    operand1
+  1  0  1    operand2
+    ----------
+  0  0  1    (operand1 & operand2) - returned result
+</code>
+In Ocrobot, the type int is a 16-bit value, so using & between two int expressions causes 16 simultaneous AND operations to occur. In a code fragment like:
+<code cpp>
+    int a =  92;    // in binary: 0000000001011100
+    int b = 101;    // in binary: 0000000001100101
+    int c = a & b;  // result:    0000000001000100, or 68 in decimal.
+</code>
+Each of the 16 bits in a and b are processed by using the bitwise AND, and all 16 resulting bits are stored in c, resulting in the value 01000100 in binary, which is 68 in decimal.
+One of the most common uses of bitwise AND is to select a particular bit (or bits) from an integer value, often called masking. See below for an example
+===== Bitwise OR (|) =====
+The bitwise OR operator in C++ is the vertical bar symbol, |. Like the & operator, | operates independently each bit in its two surrounding integer expressions, but what it does is different (of course). The bitwise OR of two bits is 1 if either or both of the input bits is 1, otherwise it is 0. In other words:
+<code cpp>
+  0  1  1    operand1
+  1  0  1    operand2
+    ----------
+  1  1  1    (operand1 | operand2) - returned result
+<
+Here is an example of the bitwise OR used in a snippet of C++ code:
+<code cpp>
+    int a =  92;    // in binary: 0000000001011100
+    int b = 101;    // in binary: 0000000001100101
+    int c = a | b;  // result:    0000000001111101, or 125 in decimal.
+</code>
+===== Example Program =====
+A common job for the bitwise AND and OR operators is what programmers call Read-Modify-Write on a port. On microcontrollers, a port is an 8 bit number that represents something about the condition of the pins. Writing to a port controls all of the pins at once.
+PORTD is a built-in constant that refers to the output states of digital pins 0,1,2,3,4,5,6,7. If there is 1 in an bit position, then that pin is HIGH. (The pins already need to be set to outputs with the pinMode() command.) So if we write PORTD = B00110001; we have made pins 2,3 & 7 HIGH. One slight hitch here is that we may also have changeed the state of Pins 0 & 1, which are used by the Ocrobot for serial communications so we may have interfered with serial communication.
+     Our algorithm for the program is:
+  * Get PORTD and clear out only the bits corresponding to the pins we wish to control (with bitwise AND).
+  * Combine the modified PORTD value with the new value for the pins under control (with biwise OR).
+<code cpp>
+int i;     // counter variable
+int j;
+void setup(){
+DDRD = DDRD | B11111100; // set direction bits for pins 2 to 7, leave 0 and 1 untouched (xx | 00 == xx)
+// same as pinMode(pin, OUTPUT) for pins 2 to 7
+Serial.begin(9600);
+}
+void loop(){
+for (i=0; i<64; i++){
+PORTD = PORTD & B00000011;  // clear out bits 2 - 7, leave pins 0 and 1 untouched (xx & 11 == xx)
+j = (i << 2);               // shift variable up to pins 2 - 7 - to avoid pins 0 and 1
+PORTD = PORTD | j;          // combine the port information with the new information for LED pins
+Serial.println(PORTD, BIN); // debug to show masking
+delay(100);
+   }
+}
+</code>
+===== Bitwise XOR (^) =====
+There is a somewhat unusual operator in C++ called bitwise EXCLUSIVE OR, also known as bitwise XOR. (In English this is usually pronounced "eks-or".) The bitwise XOR operator is written using the caret symbol ^. This operator is very similar to the bitwise OR operator |, only it evaluates to 0 for a given bit position when both of the input bits for that position are 1:
+<code cpp>
+  0  1  1    operand1
+  1  0  1    operand2
+    ----------
+  1  1  0    (operand1 ^ operand2) - returned result
+</code>
+Another way to look at bitwise XOR is that each bit in the result is a 1 if the input bits are different, or 0 if they are the same.
+Here is a simple code example:
+<code cpp>
+    int x = 12;     // binary: 1100
+    int y = 10;     // binary: 1010
+    int z = x ^ y;  // binary: 0110, or decimal 6
+</code>
+The ^ operator is often used to toggle (i.e. change from 0 to 1, or 1 to 0) some of the bits in an integer expression. In a bitwise OR operation if there is a 1 in the mask bit, that bit is inverted; if there is a 0, the bit is not inverted and stays the same. Below is a program to blink digital pin 5.
+<code cpp>
+// Blink_Pin_5
+// demo for Exclusive OR
+void setup(){
+DDRD = DDRD | B00100000; // set digital pin five as OUTPUT
+Serial.begin(9600);
+}
+void loop(){
+PORTD = PORTD ^ B00100000;  // invert bit 5 (digital pin 5), leave others untouched
+delay(100);
+}
+</code>

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