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.
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Here are the solutions to the problems: Question 1: Given the sets: P = \..., -6, -4, -2, 0, 2, 4, 6, ...\ (Set of all even integers) Q = \x: -7 < x < 7, x Z\ = \-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6\ R = \x: x > 0, x Z\ = \1, 2, 3, 4, 5, 6, 7, ...\ (Set of all positive integers) Universal set = \x: x Z\ (Set of all integers) a) Find P R: Step 1: Identify the elements common to both set P (even integers) and set R (positive integers). P R = \x : x is an even integer and x > 0\ P R = \2, 4, 6, 8, ...\ The intersection is the set of all positive even integers. P R = \2, 4, 6, 8, ...\ b) Find P' Q: Step 1: Determine the complement of P, denoted as P'. Since P is the set of all even integers and the universal set is all integers, P' is the set of all odd integers. P' = \..., -5, -3, -1, 1, 3, 5, ...\ Step 2: Find the intersection of P' and Q. P' Q = \x : x is an odd integer and -7 < x < 7\ P' Q = \-5, -3, -1, 1, 3, 5\ P' Q = \-5, -3, -1, 1, 3, 5\ c) Find n(Q R): Step 1: Find the intersection of set Q and set R. Q R = \x : -7 < x < 7 and x > 0, x Z\ Q R = \1, 2, 3, 4, 5, 6\ Step 2: Count the number of elements in the set Q R. n(Q R) = 6 n(Q R) = 6 Question 2: A binary operation is defined on the set of real numbers, R, by p q = (3p + 2q)/(2p + 3q). a) Evaluate (sqrt(3)) * (sqrt(2)). Step 1: Substitute p = sqrt(3) and q = sqrt(2) into the given formula. (sqrt(3)) * (sqrt(2)) = 3(sqrt(3)) + 2(sqrt(2))2(sqrt(3)) + 3(sqrt(2)) = 3sqrt(3) + 2sqrt(2)2sqrt(3) + 3sqrt(2) Step 2: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 2sqrt(3) - 3sqrt(2). 3sqrt(3) + 2sqrt(2)2sqrt(3) + 3sqrt(2) × 2sqrt(3) - 3sqrt(2)2sqrt(3) - 3sqrt(2) Step 3: Expand the numerator and the denominator. Numerator: (3sqrt(3) + 2sqrt(2))(2sqrt(3) - 3sqrt(2)) = (3sqrt(3))(2sqrt(3)) - (3sqrt(3))(3sqrt(2)) + (2sqrt(2))(2sqrt(3)) - (2sqrt(2))(3sqrt(2)) = 6(3) - 9sqrt(6) + 4sqrt(6) - 6(2) = 18 - 9sqrt(6) + 4sqrt(6) - 12 = 6 - 5sqrt(6) Denominator (using (a+b)(a-b) = a^2 - b^2): (2sqrt(3) + 3sqrt(2))(2sqrt(3) - 3sqrt(2)) = (2sqrt(3))^2 - (3sqrt(2))^2 = (4 × 3) - (9 × 2) = 12 - 18 = -6 Step 4: Combine the simplified numerator and denominator. 6 - 5sqrt(6)-6 Step 5: Simplify the expression. (6)/(-6) - 5sqrt(6)-6 = -1 + 5sqrt(6)6 -1 + 5sqrt(6)6 Drop the next question!