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: Sum the number of elements in all regions of the Venn Diagram to find n(P Q R). n(P Q R) = (16-x) + (6+x) + (19-3x) + 5x + 8x + 7x + 4x Combine the constant terms: 16 + 6 + 19 = 41 Combine the x terms: -x + x - 3x + 5x + 8x + 7x + 4x = (-1+1-3+5+8+7+4)x = 21x So, the total number of elements in the union of P, Q, and R is: n(P Q R) = 41 + 21x Step 2: Use the given total number of elements in the universal set, n(U) = 125, to find the value of x. Since no elements are shown outside the sets P, Q, and R, we assume n(P Q R) = n(U). 41 + 21x = 125 Subtract 41 from both sides: 21x = 125 - 41 21x = 84 Divide by 21: x = (84)/(21) x = 4 The value of x is 4. Step 3: Calculate n(P Q R)'. This represents the number of elements in the universal set that are not in P, Q, or R. n(P Q R)' = n(U) - n(P Q R) Substitute the value of x=4 into the expression for n(P Q R): n(P Q R) = 41 + 21(4) n(P Q R) = 41 + 84 n(P Q R) = 125 Now, calculate n(P Q R)': n(P Q R)' = 125 - 125 n(P Q R)' = 0 The value of n(P Q R)' is 0. That's 2 down. 3 left today — send the next one.