2015 OG Quant Review PS #18. If X and Y are sets of integers, X∆Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 6 of the integers are in both X and Y, then X∆Y consists of how many integers?
Math Lessons: (1) Learn the 2×2 chart. Speak the rows: X, ∼X (not X), and Σ (total). Speak the columns: Y, ~Y (not Y), and Σ (total). Then, seek to understand these four cells within the chart: X & Y (“Both”), X & ∼Y (“X only”), ∼X & Y (“Y only”), and ∼X & ∼Y (“Neither”). The sums to the right and to the bottom are what you’d expect; for instance, the X row and Σ column means the sum of all elements in X, whether they are in Y or not. Similarly, the Σ row and ∼Y column means the sum of all elements satisfying ∼Y, whether they are in X or not. Note that we can fill out the four remaining cells of this chart above with 0 in “Neither”, 12 in the Σ of ∼X, 4 in the Σ of ∼Y, and 22 in the Total. Fill in these values yourself to see that they make sense; (2) X∆Y is called the Symmetric Difference. The symmetric difference, X∆Y, is defined as the Union, X∪Y, less the Intersection, X∩Y. The Union of X and Y, X∪Y, is defined as X or Y or both – like a union of states. The Intersection of X and Y, X∩Y, is defined as both – like an intersection of two streets; (3) Notice that the conjunction ‘or’ leads to a Union and that the conjunction ‘and’ leads to an Intersection. For example, muscles or brown hair leads to three possibilities: (i) muscles and brown hair, (ii) muscles and ∼brown hair, & (iii) ∼muscles and brown hair. You only need to meet one of the two conditions. Meanwhile muscles and brown hair allows only one possibility: (i) muscles and brown hair; (4) Knowing ‘and’ versus ‘or’ helps you translate from English to Algebra; (5) You should reread this page; and (6) Each question below can be solved with a 2×2 chart!
Character count: The OG’s solution uses 287 characters; Shawn Berry’s solution uses 85 characters. The OG solution uses 338% as many characters yet shows 4 data points, not 9. Each line matters; each character matters.
Shawn Berry (650 level) M∆N denotes the set of elements in set M or set N, but not both. M∪N denotes the set of elements in set M or set N or both. M∩N denotes the set of elements in sets M and N. Find the probability of M∆N, P(M∆N).
A. P(M∆N) = P(M∪N) + P(M∩N)
B. P(M∆N) = P(M∪N) – P(M∩N)
C. P(M∆N) = [P(M∪N) + P(M∩N)] / 2
D. P(M∆N) = [P(M∪N) – P(M∩N)] / 2
E. P(M∆N) = P(M∪N) * P(M∩N)
Shawn Berry (700 level). Let #(X) denote the number of elements in set X.
Find #(A∩B), the number of elements in the intersection of sets A and B.
A. #(A∩B) = #(A) + #(B)
B. #(A∩B) = #(A) * #(B)
C. #(A∩B) = [#(A) + #(B)] / 2
D. #(A∩B) = #(A) + #(B) – #(A∪B)
E. #(A∩B) = [#(A) + #(B) + #(A∪B)] / 2
Shawn Berry (750 level). Determine the probability of the intersection of ~E and ~F, i.e. P(~E∩~F).
A. P(~E∩~F) = P(~E) * P(~F)
B. P(~E∩~F) = P(~E) + P(~F)
C. P(~E∩~F) = P(~E) + P(~F) – P(~E∪~F)
D. P(~E∩~F) = P(~E) + P(~F) – P(~E∩~F)
E. P(~E∩~F) = P(~E) + P(~F) – P(~E) * P(~F)
Shawn Berry (800 level). Determine the number of elements in the union of X and Y, #(X∪Y), by using the symmetric difference of X and Y, X∆Y, where the symmetric difference is the union less the intersection.
A. #(X∪Y) = #(X) + #(Y)
B. #(X∪Y) = #(X) + #(Y) + #(X∆Y)
C. #(X∪Y) = [#(X) + #(Y) + #(X∆Y)] / 2
D. #(X∪Y) = [#(X) * #(Y) + #(X∆Y)] / 2
E. #(X∪Y) = #(X) * #(Y) – #(X∆Y)
Legal Note: “The Graduate Management Admissions Council (GMAC) firmly believes that the Official Guide for GMAT Review is all that you need to perform your best on the GMAT … and that no additional techniques or strategies are needed to do well.” I, Shawn Berry, know better. I have twice earned a perfect 800 on the GMAT-CAT. I document that the Official Guide writes inconsistent, inefficient, and downright confusing solutions that take longer than the allotted 2 minutes/question. Herein I make fair use of GMAC copyrighted material – mostly its confusing solutions – for the transformative educational purpose of teaching students the clear, consistent, and efficient Mathematics, Grammar, and Logic needed to answer GMAT questions in less than 2 minutes.