XOR of three values

Question

What is the simplest way to do a three-way exclusive OR?

In other words, I have three values, and I want a statement that evaluates to true IFF only one of the three values is true.

So far, this is what I've come up with:

((a ^ b) && (a ^ c) && !(b && c)) || ((b ^ a) && (b ^ c) && !(a && c)) || ((c ^ a) && (c ^ b) && !(a && b))

Is there something simpler to do the same thing?

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Here's the proof that the above accomplishes the task:

a = true; b = true; c = true
((a ^ b) && (a ^ c) && !(b && c)) || ((b ^ a) && (b ^ c) && !(a && c)) || ((c ^ a) && (c ^ b) && !(a && b))
=> false

a = true; b = true; c = false
((a ^ b) && (a ^ c) && !(b && c)) || ((b ^ a) && (b ^ c) && !(a && c)) || ((c ^ a) && (c ^ b) && !(a && b))
=> false

a = true; b = false; c = true
((a ^ b) && (a ^ c) && !(b && c)) || ((b ^ a) && (b ^ c) && !(a && c)) || ((c ^ a) && (c ^ b) && !(a && b))
=> false

a = true; b = false; c = false
((a ^ b) && (a ^ c) && !(b && c)) || ((b ^ a) && (b ^ c) && !(a && c)) || ((c ^ a) && (c ^ b) && !(a && b))
=> true

a = false; b = true; c = true
((a ^ b) && (a ^ c) && !(b && c)) || ((b ^ a) && (b ^ c) && !(a && c)) || ((c ^ a) && (c ^ b) && !(a && b))
=> false

a = false; b = true; c = false
((a ^ b) && (a ^ c) && !(b && c)) || ((b ^ a) && (b ^ c) && !(a && c)) || ((c ^ a) && (c ^ b) && !(a && b))
=> true

a = false; b = false; c = true
((a ^ b) && (a ^ c) && !(b && c)) || ((b ^ a) && (b ^ c) && !(a && c)) || ((c ^ a) && (c ^ b) && !(a && b))
=> true

a = false; b = false; c = false
((a ^ b) && (a ^ c) && !(b && c)) || ((b ^ a) && (b ^ c) && !(a && c)) || ((c ^ a) && (c ^ b) && !(a && b))
=> false

Answer 1

For exactly three terms, you can use this expression:

(a ^ b ^ c) && !(a && b && c)

The first part is true iff one or three of the terms are true. The second part of the expression ensures that not all three are true.

Note that the above expression does NOT generalize to more terms. A more general solution is to actually count how many terms are true, so something like this:

int trueCount =
   (a ? 1 : 0) +
   (b ? 1 : 0) +
   (c ? 1 : 0) +
   ... // more terms as necessary 

return (trueCount == 1); // or some range check expression etc

Answer 2

bool result = (a?1:0)+(b?1:0)+(c?1:0) == 1;

Answer 3

a^b^c is only 1 if an uneven number of variables is 1 (two '1' would cancel each other out). So you just need to check for the case "all three are 1":

result = (a^b^c) && !(a&&b&&c)

Answer 4

Another possibility:

a ? !b && !c : b ^ c

which happens to be 9 characters shorter than the accepted answer :)

Answer 5

You could also try (in C):

!!a + !!b + !!c == 1

Answer 6

Here's a general implementation that fails quickly when more than one bool is found to be true.

Usage:

XOR(a, b, c);

Code:

public static bool XOR(params bool[] bools)
{
    return bools.Where(b => b).AssertCount(1);
}

public static bool AssertCount<T>(this IEnumerable<T> source, int countToAssert)
{
    int count = 0;
    foreach (var t in source)
    {
        if (++count > countToAssert) return false;
    }

    return count == countToAssert;
}

Answer 7

f= lambda{ |a| [false, false, true].permutation.to_a.uniq.include? a }
p f.call([false, true, false])
p f.call([false, true, true])

$ true

$ false

Because I can.

Answer 8

Better yet on Python:

result = (1 if a else 0)+(1 if b else 0)+(1 if c else 0) == 1

This can be used also on if statements!

It saved my day for CLI mutually exclusive arguments through Click (everyone hates click)