In a class of 177 students, 63 of them were born in months that include at least one u in their spelling (January, February, June, July, August), 112 were born in moths with at least one r in their spelling (January, February, March, April, September, October, November, December) , 71 were born in months with at least one m AND at least one r in their spelling (March, September, November, December) and 23 were born in the first two months of the year. How many were born in May? Can you check my answer: Solution. (a) Let A be the set of students born in months that contain at least one u (January, February, June, July, August), with |A| = 63. (b) Let B be the set of students born in months that contain at least one r (January, February, March, April, September, October, November, December), with |B| = 112. (c) Let C be the set of students born in months that contain at least m AND at least one r (March, September, November, December), with |C| = 71. (d) Let D be the set of students born in the first two months of the year (January, February), with |D| = 23. Since C ⊆ B and only students who born in May remains the outside all these sets, we can apply the Principle of Inclusion-Exclusion (PIE) for set A, B, and D. |A ∪ B ∪ D| = |A| + |B| + |D| − |A ∩ B| − |B ∩ D| − |A ∩ D| + |A ∩ B ∩ D| (a) |A ∩ B| = 23 because A ∩ B represents students born in months with both u and r, which are January and February, covered by set D. (b) |B∩D|=23becauseD⊆B. (c) |A∩D|=23becauseD⊆A. (d) |A∩B∩D|=23becausesetDisentirelywithinsetAandsetB. Therefore, |A ∪ B ∪ D| = |A| + |B| + |D| − |A ∩ B| − |B ∩ D| − |A ∩ D| + |A ∩ B ∩ D| = 63 + 112 + 23 − 23 − 23 − 23 + 23 = 152 To find the number of students born in May, we subtract |A ∪ B ∪ D| from the total number of students: 177−152 = 25