# What does upside down V and plus inside a circle mean?

By | April 19, 2015

If you have been looking at bitcoins and in particular the SHA256 algorithm, you would have come across this formula:

## Maj(x,y,z)=(x∧y)⊕(x∧z)⊕(y∧z) Ch(x,y,z)=(x∧y)⊕(¬x∧z)

The first thing I said to myself was “What the heck does this mean? Do I need a PhD to understand this?”

Well, the good news is that the meaning is pretty simple and no you don’t need a PhD.

First of all, these are logic symbols that students studying logic use. Programmers don’t use these.

The upside down V (∧) means AND.
The circle with the plus inside (⊕) means XOR or eXclusive OR.
The bent line or garden hoe as I call it (¬) is the negation or a NOT.
Side note: V means OR.

AND is easy to understand.
0 AND 0 = 0
0 AND 1 = 0
1 AND 0 = 0
1 AND 1 = 1

ie both inputs need to be true for the output to be true.

Check out this example. 116 & 122 = 112. Why?
01110100 = 116 (t)
01111010 = 122 (z)
————–
01110000 = 112 (p)

If converted to ascii text, you get t & z = p. Cool eh?

XOR is a bit tricky. Basically the output is true when the inputs are different.
0 AND 0 = 0
0 AND 1 = 1
1 AND 0 = 1
1 AND 1 = 0

Negation is easy as well. Whatever the value is, make it the opposite.
0 => 1
1 => 0

Wikipedia (http://en.wikipedia.org/wiki/List_of_logic_symbols) has a full list of other symbols.

### How does this relate to programming?

In the C language, these are bitwise operators. I’ll explain more below.
∧ becomes &
⊕ becomes ^ (caret)
¬ becomes ~
V become | (just for completeness)

So to represent this in C, it becomes:
#define CH(x,y,z) (((x) & (y)) ^ (~(x) & (z)))

But hang on. You might be wondering what && and || and ! mean? These also mean AND, OR and NOT respectively but these are logical operators, not bitwise operators. (Why did I not mention logical XOR? Because there isn’t one. Check out: http://benpfaff.org/writings/clc/logical-xor.html and see if that makes sense).

### What’s the diff?

The logical operator works on boolean (true or false conditions) and the bitwise operator works on bits (1’s and 0’s).

### What the heck does that mean?

Check out this example for logical operators. http://www.tutorialspoint.com/cprogramming/c_logical_operators.htm

And check out this example for bitwise operators.
http://www.tutorialspoint.com/cprogramming/c_bitwise_operators.htm

You’ll notice that generally speaking you’ll not need to use bitwise operators…. unless you want to play with 1’s and 0’s.