The Toffoli gate serves as a universal gate for Boolean logic, if we can provide fixed input bits and ignore output bits. If z is initially 1, then x ↑ y = 1 − xy appears in the third output — we can perform NAND. If we fix x = 1, the Toffoli gate functions like an XOR gate, and we can use it to copy. The Toffoli gate θ (3) is universal in the sense that we can build a circuit to compute any reversible function using Toffoli gates alone (if we can fix input bits and ignore output bits).

The Toffoli gate and the negation gate together yield a universal gate set, in the sense that every permutation of {0,1}n can be implemented as a composition of these gates. Since every bit operation that does not use all of the bits performs an even permutation, we need to use at least one auxiliary bit to perform every permutation, and it is known that one bit is indeed enough. Without auxiliary bits, all even permutations can be implemented.

A gate is just a linear transformation, which means that it can be represented by a matrix. So the Toffoli gate is one 3 x 3 matrix. Therefore it acts on 3-component objects and it acts on these in a particular way: it mixes up two of them and does something else to the third. This means that it's possible to construct three Toffoli gates.

And now it's possible to show (a) that the action of these transformations is a permutation and (b) that ANY permutation of the components of the vectors on which these gates act can be represented by composing these three gates-hence the universality.

Indeed it's possible to show that the same holds for n-component vectors: That there do exist n x n matrices, that generalize for n variables what the Toffoli gate does for three variables and describe permutations.

The Toffoli gate serves as a universal gate for Boolean logic, if we can provide fixed input bits and ignore output bits. If z is initially 1, then x ↑ y = 1 − xy appears in the third output — we can perform NAND. If we fix x = 1, the Toffoli gate functions like an XOR gate, and we can use it to copy. The Toffoli gate θ (3) is universal in the sense that we can build a circuit to compute any reversible function using Toffoli gates alone (if we can fix input bits and ignore output bits).

The Toffoli gate and the negation gate together yield a universal gate set, in the sense that every permutation of {0,1}n can be implemented as a composition of these gates. Since every bit operation that does not use all of the bits performs an even permutation, we need to use at least one auxiliary bit to perform every permutation, and it is known that one bit is indeed enough. Without auxiliary bits, all even permutations can be implemented.