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 set theory problem: First, let's list the elements of the universal set , set P, and set Q. Step 1: Define the sets. The universal set = \x; 1 < x < 20, where x is an integer\. = \2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\ Set P = \x: x is a multiple of 3\. P = \3, 6, 9, 12, 15, 18\ Set Q = \x: x is a prime number\. Q = \2, 3, 5, 7, 11, 13, 17, 19\ Step 2: Find the complements P' and Q'. P' contains all elements in that are not in P. P' = \2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19\ Q' contains all elements in that are not in Q. Q' = \4, 6, 8, 9, 10, 12, 14, 15, 16, 18\ Now we can solve parts a), b), and c). a) Find P Q' This is the intersection of set P and set Q', meaning elements that are in both P and Q'. P Q' = \3, 6, 9, 12, 15, 18\ \4, 6, 8, 9, 10, 12, 14, 15, 16, 18\ P Q' = \6, 9, 12, 15, 18\ b) Find P' Q This is the union of set P' and set Q, meaning all elements that are in P' or in Q (or both). P' Q = \2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19\ \2, 3, 5, 7, 11, 13, 17, 19\ P' Q = \2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19\ c) Find (P Q)' First, find the union of P and Q. P Q = \3, 6, 9, 12, 15, 18\ \2, 3, 5, 7, 11, 13, 17, 19\ P Q = \2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19\ Next, find the complement of (P Q), which means elements in that are not in P Q. (P Q)' = (P Q) (P Q)' = \2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\ \2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19\ (P Q)' = \4, 8, 10, 14, 16\ 3 done, 2 left today. You're making progress.