4. De Morgan's Law: Let A, A2,. , be a sequence of countable events. Show that
Let A and B be events, and consider the following statement from de Morgan's Law: (A ∩ B)C = AC ∪ BC Prove this statement.
Show: Let A1, A2, ... be a infinitely countable collection of events, then P lim P (UA m+00 i=1
LIDO D EDIUL 9. Let A1, A2, ... be a sequence of events. Show that PA A - A) = P(A) - PA A - UAA-2 UAA) for i = 2, 3,.... Hint: You don't need induction to prove this. You can assume, without proof, that A A-2 UA A-2 UAA = A (A-2 UA-2 UA) and A = AB Ü AB
6. Show that if A1, A2, ... is an expanding sequence of events, that is, AC A₂C...... then P(ALU AQU....) = lim P(An). 1-00
5. Prove De Morgan's law: (An B)'« A' U B: (Don't use Venn diagram.)
Z XY +X'Y Implement this Boolean expression using only NOR gates. Apply De Morgan's law and Boolean laws for the expression to represent it only using NOR operation. Implement it using 4 NOR gates only
Problem 4. Let Ai, A2,..., An be events. Prove .+P(Ann
Q.2) Using De Morgan's law: a) Design a 3-input NOR gate using 2-input NOR gate only. Draw you diagram b) Design 4 input AND gate using 2 input NOR gates. Draw you diagram
3. Let (12, F,P) be a probability space, and A1, A2, ... be an increasing sequence of events; that is, A1 CA2 C.... Using only the Kolmogorov axioms, prove that P is continuous from belour: lim P(An) = P(U=1 An). Hint: Work with a new sequence of events By := A and B := An An-1. n+00 [1]
Please follow the recommendations suggested and use de Morgan's laws. I would like know how de Morgan's law is used to proof the theorem. (The First Partition Theorem). For any ACR, we have: (16.20) Theorem 16.1.11 Int(A) U a(A) U Ext(A) = R; Int(A) n a(A) Ø; Int(A)n Ext(A) = Ø; a(A)n Ext(A) = Ø. Proof. The proof simply boils down to writing the definitions of the sets in the right way and applying De Morgan's Laws for each of...