Sets

Set of vowels in English alphabet, Set of days in a week, Set of students in your class

Finite sets have countable elements. Vowels = {a, e, i, o, u} (5 elements), Days = {Monday, Tuesday, ...Sunday} (7 elements), Students in class (countable number). Natural numbers and whole numbers are infinite sets.

φ ⊆ A for any set A, n(φ) = 0, φ ∩ A = φ for any set A, φ ∪ A = A for any set A

Empty set φ has no elements, so n(φ) = 0. It's subset of every set. φ ∩ A = φ (no common elements), φ ∪ A = A (union adds nothing). φ has zero elements, not one.

17

Students who like only Mathematics = Total who like Math - Those who like both = 25 - 8 = 17. Using Venn diagram: Only Math = 17, Both = 8, Only Science = 12.

{2, 4, 6, 8}

A' (complement of A) contains all elements in U that are not in A. U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 3, 5, 7}. So A' = {2, 4, 6, 8}.