The empty set has no elements. A finite set has a definite number of elements; an infinite set does not. Two sets are equal when they have exactly the same elements, whatever the order or the way they are written.
A set can have no elements. It is called the empty set, or null set, and is written ∅ or { }.
The empty set appears when a rule is satisfied by no object. { x : x is a real number and x² = −1 } is empty, because the square of every real number is zero or positive. The set of months with 32 days is empty.
All empty sets are the same set. Two sets are equal when they have the same elements, and two sets with no elements have the same elements: none.
{0} is not empty. It has one element, the number 0. {∅} is not empty either. It has one element, which is the empty set.
Is { x : x is a whole number and 3 < x < 4 } empty?
Yes, it is ∅.
The rule asks for whole numbers strictly greater than 3 and strictly less than 4. The whole number after 3 is 4, which is not less than 4. So no whole number satisfies both conditions, and the set has no elements.
A finite set has a definite number of elements, which could in principle be listed completely. The set of vowels has 5 elements; the set of days of the week has 7; the empty set has 0. All are finite.
An infinite set is not finite: however many elements are listed, more remain. The set of natural numbers {1, 2, 3, …} is infinite. So is the set of even numbers. An infinite set is written in set-builder form, or in roster form with “…” when the pattern is clear.
The number of elements in a finite set A is its cardinal number, written n(A). If V is the set of vowels, n(V) = 5. n(∅) = 0.
Is each set finite or infinite: (a) the letters of the English alphabet, (b) all multiples of 5, (c) all prime numbers?
(a) Finite. There are exactly 26 letters.
(b) Infinite. For any multiple of 5, adding 5 gives a larger one.
(c) Infinite. Euclid proved this around 300 BCE. Suppose there were only finitely many primes. Multiply them all together and add 1. The result is not divisible by any prime on the list, because each leaves remainder 1. So its prime factors are primes not on the list. This contradicts the assumption that the list was complete.
Two sets A and B are equal, written A = B, when every element of A is an element of B and every element of B is an element of A. Only the elements matter, not their order or how the set is described.
{1, 2, 3} = {3, 2, 1}. { x : x is a vowel of the English alphabet } = {a, e, i, o, u}. But {1, 2, 3} ≠ {1, 2, 3, 4}, because 4 is in the second set and not in the first.
A = { x : x is a letter of the word “FOLLOW” } and B = { x : x is a letter of the word “WOLF” }. Is A = B?
Write each in roster form, each element once. A = {F, O, L, W}. B = {W, O, L, F}.
Both contain exactly F, O, L and W. So A = B, although the words have different lengths and different letter orders.
1A set with no elements is called the ____ set, written ∅ or { }.
2Which of these is the empty set?
3Finite or infinite?
4The cardinal number of {a, e, i, o, u} is…
5Two sets are equal when…
6Is {1, 2, 3} equal to {3, 1, 2}?
Next: subsets, the power set, and intervals on the number line.
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