Starting with NAND only...

logicGates.js

     var logicGates = {
        nand: function (a, b) {
                return !(a && b);
        },
        not:  function (a)    {
                return this.nand(a, a);
        },
        and:  function (a, b) {
                return this.not(this.nand(a, b));
        },
        or:   function (a, b) {
                return this.nand(this.not(a), this.not(b));
        },
        nor:  function (a, b) {
                return this.not(this.or(a, b));
        },
        xor:  function (a, b) {
                return this.and(this.nand(a, b), this.or(a, b));
        },
        xnor: function (a, b) {
                return this.not(this.xor(a, b));
        }
    };

    [
        {a: false, b: false},
        {a: false, b: true},
        {a: true, b: false},
        {a: true, b: true},
    ].map(function (x) { return logicGates.nand(x.a, x.b); });