This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's how to solve the problem: Step 1: Calculate the number of elements in each distinct region of the Venn Diagram. We start with the innermost intersection and work outwards: • n(P Q R) = 3 • n(P Q only) = n(P Q) - n(P Q R) = 7 - 3 = 4 • n(P R only) = n(P R) - n(P Q R) = 8 - 3 = 5 • n(Q R only) = n(Q R) - n(P Q R) = 5 - 3 = 2 • n(P only) = n(P) - (n(P Q only) + n(P R only) + n(P Q R)) n(P only) = 20 - (4 + 5 + 3) = 20 - 12 = 8 • n(Q only) = n(Q) - (n(P Q only) + n(Q R only) + n(P Q R)) n(Q only) = 16 - (4 + 2 + 3) = 16 - 9 = 7 • n(R only) = n(R) - (n(P R only) + n(Q R only) + n(P Q R)) n(R only) = 21 - (5 + 2 + 3) = 21 - 10 = 11 Step 2: Represent this information on a Venn Diagram (Part a). Since I cannot draw a diagram, here are the values for each region: • Elements in P only: 8 • Elements in Q only: 7 • Elements in R only: 11 • Elements in P Q only (not in R): 4 • Elements in P R only (not in Q): 5 • Elements in Q R only (not in P): 2 • Elements in P Q R: 3 Step 3: Find n(P Q R) (Part b.i). Using the Principle of Inclusion-Exclusion: n(P Q R) = n(P) + n(Q) + n(R) - n(P Q) - n(P R) - n(Q R) + n(P Q R) Substitute the given values: n(P Q R) = 20 + 16 + 21 - 7 - 8 - 5 + 3 n(P Q R) = 57 - 20 + 3 n(P Q R) = 37 + 3 n(P Q R) = 40 Step 4: Find n((P Q)' R) (for Part b.ii). The set (P Q)' R consists of all elements that are either not in P or Q, or are in R. From the Venn Diagram regions, the elements not in P or Q are those in R only (which is 11), plus any elements outside P Q R. The elements in R are n(R) = 21. The union (P Q)' R can be understood as all elements in R, plus any elements that are outside P Q R. If we assume there are no elements outside P Q R (i.e., n(U) = n(P Q R)), then the elements not in P or Q are simply the elements in R only. In this case, (P Q)' R = (R only) R = R. So, n((P Q)' R) = n(R) = 21. Step 5: Calculate the probability of (P Q)' R (Part b.ii). To calculate probability, we need a universal set n(U). Since no universal set is specified, we assume the universal set is P Q R. So, n(U) = n(P Q R) = 40. The probability is: P((P Q)' R) = (n((P Q)' R))/(n(U)) P((P Q)' R) = (21)/(40) The probability of (P Q)' R is (21)/(40). 3 done, 2 left today. You're making progress.