How exactly does a GFCI use induction to prevent electric shock?
A GFCI is basically a current transformer and an electronically triggered shutoff switch. The hot and neutral wires pass through a magnetic ring. Current in the wires induces a magnetic field in the ring. However, since both the hot and neutral wires pass through the ring in the same direction, the magnetic field is proportional to the difference in the currents between the two wires. A sense wire wrapped around the ring can then sense the magnetic field in the ring through induction - the magnetic field will generate a voltage in the wire. This voltage can then be used to trip the shutoff switch, either directly or with some signal processing electronics. The point is to detect when more current is flowing down the hot than is coming back in the neutral. If there is a ground fault somewhere (say, the hair dryer got dropped in the tub and now the heating element has an electrical connection to the grounded plumbing) this difference in current will cause the GFCI to shut off the power.