Sets

Equivalent sets

Step 1: We need to compare sets A = {1, 2, 3} and B = {a, b, c}. Step 2: Equal sets have exactly the same elements. A and B have different elements (numbers vs letters), so they are NOT equal sets. Step 3: Equivalent sets have the same NUMBER of elements (elements can be different). Step 4: n(A) = 3 and n(B) = 3 — both sets have 3 elements each. Step 5: Since the number of elements is equal (but elements themselves are different), A and B are equivalent sets. Also note they are disjoint (no common element), but the best description is 'equivalent sets' as it captures the key relationship.

8

Step 1: Find the set of letters used in the word MATHEMATICS. Step 2: List all letters: M, A, T, H, E, M, A, T, I, C, S. Step 3: Remove repeated letters — M appears twice, A appears twice, T appears twice. Step 4: Unique letters are: M, A, T, H, E, I, C, S. Step 5: Count the unique letters: M, A, T, H, E, I, C, S = 8 letters. So n(B) = 8. A common mistake is to count all 11 letters in the word without removing repetitions.

Set of odd natural numbers

Step 1: An infinite set has an unlimited number of elements that cannot be fully counted. Step 2: 'Natural numbers less than 100' = {1, 2, 3, ..., 99} — this has exactly 99 elements, so it is FINITE. Step 3: 'Students in a school' is a limited, countable number — so FINITE. Step 4: 'Days in a week' = {Monday, Tuesday, ..., Sunday} — only 7 elements, so FINITE. Step 5: 'Odd natural numbers' = {1, 3, 5, 7, 9, ...} — the dots (…) show it goes on forever with no end. This is an INFINITE set.

{ }

The empty set has no elements. In the chapter, the empty set is shown by { } or by the symbol ∅. So the set with no elements is the null set.