a.) AUB represents the set of numbers that are either odd or less than 50. So, it includes all odd numbers less than 50 (A) as well as all numbers less than 50 (U). Therefore, AUB is the set of all numbers less than 50.
b.) AUC represents the set of numbers that are either odd or prime or less than 50. So, it includes all odd numbers less than 50 (A), all prime numbers less than 50 (C), and all numbers less than 50 (U). Therefore, AUC is the set of all numbers less than 50.
c.) (A-C)' represents the set of numbers that are in A but not in C, and then takes the complement of that set. A is the set of odd numbers less than 50, and C is the set of prime numbers less than 50. The numbers that are in A but not in C are the odd composite numbers less than 50. Taking the complement of that set means including all the numbers that are not in the set. Therefore, (A-C)' is the set of all even numbers less than 50.
d.) (BUC) represents the set of numbers that are either even or prime or less than 50. So, it includes all even numbers less than 50 (B), all prime numbers less than 50 (C), and all numbers less than 50 (U). Therefore, (BUC) is the set of all numbers less than 50.
e.) A D represents the intersection of sets A and D. A is the set of odd numbers less than 50, and D is the set of multiples of 3 less than 50. The intersection of these two sets is the set of odd multiples of 3 less than 50.
PH