L1 · Unions, Intersections and Venn Diagrams
“I can build the union and intersection of two events as sets of outcomes, read and complete a Venn diagram from region counts, and tell whether two events are mutually exclusive.”
One roll of a die. Let A = even = {2, 4, 6} and B = greater than 3 = {4, 5, 6}. Step 1: The intersection collects the outcomes in BOTH sets: A ∩ B = {4, 6}. Step 2: The union collects every outcome in either set - each one listed once, even if it appears in both: A ∪ B = {2, 4, 5, 6}. Step 3: Draw the Venn in words: A-only region {2} (1 outcome); overlap {4, 6} (2 outcomes); B-only region {5} (1 outcome); outside both circles {1, 3} (2 outcomes). Regions: 1 + 2 + 1 + 2 = 6 - the whole sample space, accounted for exactly once. Step 4: Probabilities are counts over the sample space: P(A ∩ B) = 2/6 = 1/3; P(A ∪ B) = 4/6 = 2/3. Say it plainly: "∩" is a filter that keeps only the overlap; "∪" is a merger that refuses to count anything twice.
The outcome 4 belongs to both A and B. In the union A ∪ B, how many times is 4 listed, and why?
Same die. Let C = multiples of 3 = {3, 6} and D = less than 5 = {1, 2, 3, 4}. List C ∩ D and C ∪ D, and find P(C ∩ D). -> C ∩ D = {3}; C ∪ D = {1, 2, 3, 4, 6}; P(C ∩ D) = 1/6
"E = {1, 2, 3} (less than 4) and F = {2, 4, 6} (even). E ∩ F = {___}, and E ∪ F = {___, ___, ___, ___, ___}." -> E ∩ F = {2}; E ∪ F = {1, 2, 3, 4, 6}
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.