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Number of Elements

Counting how many elements are in a set, and in combinations of sets.

Counting Elements

When two sets overlap, simply adding n (A) + n (B) counts the shared elements twice. To fix this, subtract the overlap once: n (A \cup B) = n (A) + n (B) - n (A \cap B) Rearranged (useful in exams): n (A \cap B) = n (A) + n (B) - n (A \cup B)

Example 1 — easy numbers

Let A = {1, 2, 3, 4} and B = {3, 4, 5}. Find n (A \cup B). 1 n (A) = 4, n (B) = 3 2 n (A \cap B): the shared elements are 3 and 4, so n (A \cap B) = 2 3 n (A \cup B) = n (A) + n (B) - n (A \cap B) = 4 + 3 - 2 = 5 Check by listing: A \cup B = {1, 2, 3, 4, 5}, which has 5 elements. ✓ Answer: 5

Example 2 — a word problem

In a class of 30 students, 18 play football, 15 play cricket, and 8 play both. How many play football or cricket? 1 Let A = football players, n (A) = 18 . Let B = cricket players, n (B) = 15 . 2 The overlap is n (A \cap B) = 8 (play both). 3 n (A \cup B) = 18 + 15 - 8 = 25 Answer: 25 students play football or cricket (or both)

Example 3 — finding students who play neither

Using the class of 30 from Example 2, how many students play neither sport? 1 From Example 2, 25 students play at least one sport. 2 Total students: 30 . Students playing neither: 30 - 25 = 5 Answer: 5 students play neither sport

Practice

Swipe or use the arrows to move between questions.

1 / 2

n (A) = 12, n (B) = 9, n (A \cap B) = 4 . What is n (A \cup B)?

Your answer

n (A) = 10, n (B) = 15, n (A \cup B) = 20 . Find n (A \cap B).

Your answer

Counting Elements

When two sets overlap, simply adding n (A) + n (B) counts the shared elements twice. To fix this, subtract the overlap once: n (A \cup B) = n (A) + n (B) - n (A \cap B) Rearranged (useful in exams): n (A \cap B) = n (A) + n (B) - n (A \cup B)

Example 1 — easy numbers

Let A = {1, 2, 3, 4} and B = {3, 4, 5}. Find n (A \cup B). 1 n (A) = 4, n (B) = 3 2 n (A \cap B): the shared elements are 3 and 4, so n (A \cap B) = 2 3 n (A \cup B) = n (A) + n (B) - n (A \cap B) = 4 + 3 - 2 = 5 Check by listing: A \cup B = {1, 2, 3, 4, 5}, which has 5 elements. ✓ Answer: 5

Example 2 — a word problem

In a class of 30 students, 18 play football, 15 play cricket, and 8 play both. How many play football or cricket? 1 Let A = football players, n (A) = 18 . Let B = cricket players, n (B) = 15 . 2 The overlap is n (A \cap B) = 8 (play both). 3 n (A \cup B) = 18 + 15 - 8 = 25 Answer: 25 students play football or cricket (or both)

Example 3 — finding students who play neither

Using the class of 30 from Example 2, how many students play neither sport? 1 From Example 2, 25 students play at least one sport. 2 Total students: 30 . Students playing neither: 30 - 25 = 5 Answer: 5 students play neither sport

Practice

Swipe or use the arrows to move between questions.

1 / 2

n (A) = 12, n (B) = 9, n (A \cap B) = 4 . What is n (A \cup B)?

Your answer

n (A) = 10, n (B) = 15, n (A \cup B) = 20 . Find n (A \cap B).

Your answer