This problem is solved on the basis of Venn Diagram:-
2.1. Use Venn diagrams to verify that (a) (A UB)UC is the same event as AU(BUC)...
Exercise 4. By writing AU BUC as (AUB) UC, show that the Principle of Inclusion-Exclusion for three sets is P(AUBUC) = P(A)+P(B)+P(C)- P(ANB) - P(ANC) - P(BNC)+P(ANBNC) Can you generalize the result to an arbitrary number of events?
2.4. Use Venn diagrams to verify that if A is contained in B, then AnB A and AnB'.
4. On the following Venn diagrams, use shaded area to represent (a) (An B')UC (b) (A -B)nc. (c) (BUC)n(A'UB). B S
1. Let A, B and C be events in the sample space S. Use Venn Diagrams to shade the areas representing the following events (32 points) a. AU (ANB) b. (ANB) U ( AB) C. AU ( ANB) d. (AUB) N (AUC)
3. (12') Using Venn diagrams, verify the following identities. (a) A-(AnB)U(A-B) ( b) If A and B are finite sets, we have (AUB)- A+B-(AnB )
Use Venn diagrams to prove or disprove the following c) AU B (An B) u (A n B)u (A n B) d) A U (B n C) (AU B) n (AU C)
Shade the appropriate areas for the event listed 13. (A UB) n(AUC)n (BUC) 14. (An B) U (A n C) U (B n c) 15. AnBnC
Please help me prove 2,4, and 5. Thank you Theorem 17. Let A, B and C be sets. Then the following statements are true: (1) AB CA; (2) B CAUB; (3) A CAUB; (4) AB=BA; (5) AU (AUC) = (AUB) UC; (6) An(BNC) = (ANB) nC; (7) An (BUC) = (ANB) U (ANC); (8) AU (BAC) = (AUB) n(AUC).
By only use these axioms to solves the following two questions. Thank you. (AUB)A nB (AnB) AUB 0 EPCA)E P(S)=I PCAUB) P(A) P(B)-PIAne) P(AIB) # ot times A and Boccur #ot times B ocuuts P(ADP(ANB) PCB) P/AB)P(BIA)P(A) P(B) Taew ledr- Using notin The defa P (A I8), The 3 axioms, and T "lews" Teem we have discussed (e. more 1 Show TR P(ALB ) PLACIB) uw-leuti9-2AsSsume AnBUc) (AnB) U (ANC) Mew show Tt Pl(AU B)UC) PIAT+ PCB) Pc) PCANB) PIANC)-...
2.2. Use Venn diagrams to verify the two De Morgan laws: (b) (AUB) A'n B'.