Finite vs Infinite Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Classify each set as finite or infinite and explain: (a) the days of the week, (b) the multiples of 77, (c) the letters in the word 'MATH', (d) the decimal digits.

Solution

  1. 1
    (a) Days of the week: {\{Mon, Tue, Wed, Thu, Fri, Sat, Sun}\}. Exactly 77 elements. Finite.
  2. 2
    (b) Multiples of 77: 7,14,21,28,…7, 14, 21, 28, \ldots There is no largest multiple (given any multiple 7n7n, the next is 7(n+1)7(n+1)). Infinite.
  3. 3
    (c) Letters in 'MATH': {M,A,T,H}\{M, A, T, H\}. Exactly 44 elements. Finite.
  4. 4
    (d) Decimal digits: {0,1,2,…,9}\{0, 1, 2, \ldots, 9\}. Exactly 1010 elements. Finite.

Answer

(a) Finite; (b) Infinite; (c) Finite; (d) Finite.
A set is finite if you can count all its elements and reach a final count. It is infinite if, no matter how many elements you list, there are always more. The multiples of any integer form an infinite set because multiplication produces new values without bound.

About Finite vs Infinite

Finite describes a quantity or set with a definite end; infinite describes something that goes on forever without bound.

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