Proofs, Languages and Grammars Two general notes about the assignments in this course... Unless noted, a. When you are asked to draw a transition graph, you do not have to draw those transitions that are not possible in the context of the problem or lead to a trap state. However, for transition matrices, always include the trap state if there is one and call it T. b. To draw the automata you can either draw them by hand in your solution paper or, preferably, draw them in JFLAP, take a screenshot and include them in your submission file (similar to the one in the problem below). 1. [20 points] Occasionally, we need to use the union and intersection symbols in a manner analogous to the summation sign . We define i=1nSi=S1S2Sn with an analogous notation for the intersection of several sets. With this notation, the general DeMorgan's laws are written as i=1nSi=i=1nSi and i=1nSi=i=1nSi Prove these identities for all finite values of n. Hint: prove it by induction on " n