Problem #$6(15$ points$)$

A PLC safety system is used for a milling machining center. When a machine tool is about to start its machining operation you need to make sure the workpiece is loaded and the safety guard is on$.$ If the safety guard switch $($X$1)$ has not been activated and the limit switch $($X$2)$ signaling the presence of the workpiece has not been activated, a warning light $(Y)$ will turn on$.$ If only one of them activated, the warning light still turns on$.$

a$.$ Construct the truth table for this system.

b$.$ Write the Boolean logic expression for the system.

c$.$ Construct the ladder logic diagram for the system.