Sets, Elements, Subsets, and Roster Form - Sixth Grade Mathematics
A set is a well-defined collection of distinct mathematical objects, called elements or members, that share specified criteria for membership.
What is a Well-Defined Set?
In mathematics, a collection is only considered a valid set if it is well-defined, meaning there is zero ambiguity or subjective opinion regarding whether any given object belongs to the collection.
Well-Defined vs. Not Well-Defined:
- Well-Defined: "The set of whole numbers less than 10."
(Membership is clear and objective: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}).
- Not Well-Defined: "The set of large numbers."
(Subjective: Is 50 large? Is 1,000,000 large? Depends on opinion).
- Well-Defined: "The set of vowels in the English alphabet."
(Strictly defined: {a, e, i, o, u}).
- Not Well-Defined: "The set of interesting books."
(Completely subjective).
Sets are traditionally denoted by capital letters, such as set A, set B, or set S.
Elements and Set Membership Notation
The individual items inside a set are called elements or members.
Standard Set Notation Symbols:
+----------------+-------------------------------+------------------------------------+
| Symbol | Meaning | Example |
+----------------+-------------------------------+------------------------------------+
| in | is an element of | 3 in A (3 is an element of set A) |
+----------------+-------------------------------+------------------------------------+
| not in | is NOT an element of | 8 not in A |
+----------------+-------------------------------+------------------------------------+
| { } or O/ | Empty Set (Null Set) | A set containing zero elements |
+----------------+-------------------------------+------------------------------------+
| U | Universal Set | The set containing all objects |
| | | under consideration |
+----------------+-------------------------------+------------------------------------+
| n(A) or |A| | Cardinality | Number of distinct elements in A |
+----------------+-------------------------------+------------------------------------+
Methods of Describing Sets
There are three primary ways to define and represent a set:
1. Roster Form (Listing Method):
Enclosing every individual element inside curly braces {}, separated by commas.
- Elements are listed without repetition!
- Order does not matter: {1, 2, 3} is identical to {3, 1, 2}.
Example: The set of prime numbers less than 12:
P = {2, 3, 5, 7, 11}.
2. Word Description:
A clear, descriptive phrase defining membership criteria.
Example: "Let E be the set of even whole numbers between 1 and 9."
3. Set-Builder Notation:
A symbolic rule format: {x | condition on x}, read as "the set of all x such that..."
Example: E = {x | x is an even integer and 1 < x < 9} = {2, 4, 6, 8}.
The empty set (or null set) contains no elements whatsoever. It is written as {} or the symbol $\emptyset$. Note: writing {\emptyset} is incorrect, as that would be a set containing the symbol as an element!
Finite, Infinite, and Universal Sets
A set is finite if its elements can be completely counted with a natural number, such as the letters in the word "MATH": $M = {M, A, T, H}$ with cardinality $n(M) = 4$. A set is infinite if the counting of elements never terminates. Ellipses (...) are used to indicate endless continuation, such as the natural numbers: $\mathbb{N} = {1, 2, 3, 4, 5, ...}$.
Subsets and Equal Sets
Two sets are equal ($A = B$) if and only if they contain the exact same elements.
Set A is a subset of Set B (written $A \subseteq B$) if every element of A is also an element of B. If A is a subset of B, but B contains at least one element not in A, then A is a proper subset ($A \subset B$).
Subsets Illustrated:
Let Universal Set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Let Set B = {2, 4, 6, 8, 10} (Even numbers)
Let Set A = {2, 4, 6}
Since every element in A is also inside B, A is a subset of B (A subset B).
Important Subset Axioms:
1. The empty set is a subset of EVERY set ($\emptyset \subseteq A$).
2. Every set is a subset of itself ($A \subseteq A$).
A set with $n$ elements has exactly $2^n$ total possible subsets.
For example, a set with 3 elements ${a, b, c}$ has $2^3 = 8$ subsets:
$\emptyset, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}$.
Exercises
Exercise 1. State whether each collection is well-defined: (a) The set of US states bordering the Pacific Ocean; (b) The set of fun video games; (c) The set of multiples of 5 less than 35.
Exercise 2. Write each set in roster form and state its cardinality $n(A)$: (a) Set of vowels in the word "GEOMETRY"; (b) Set of whole numbers between 12 and 18; (c) Set of two-digit numbers whose digits sum to 5.
Exercise 3. Let $S = {1, 3, 5, 7, 9}$. True or False: (a) $3 \in S$; (b) $4 \in S$; (c) ${5, 7} \subseteq S$; (d) ${0} \subseteq S$; (e) $\emptyset \subseteq S$.
Exercise 4. List all the subsets of the set $T = {x, y}$. How many total subsets are there?
Solutions
Solution to Exercise 1. For (a), well-defined because geographic state borders are established legal boundaries (Washington, Oregon, California, Alaska, Hawaii). For (b), not well-defined because "fun" is a subjective personal opinion. For (c), well-defined because multiples of 5 under 35 are precisely ${5, 10, 15, 20, 25, 30}$.
Solution to Exercise 2. For (a), the distinct vowels in "GEOMETRY" are E and O. In roster form: ${E, O}$. Cardinality $n(A) = 2$ (the second E is not repeated). For (b), numbers strictly between 12 and 18: ${13, 14, 15, 16, 17}$. Cardinality $n(B) = 5$. For (c), two-digit numbers whose digits add to 5 are 14, 23, 32, 41, 50. In roster form: ${14, 23, 32, 41, 50}$. Cardinality $n(C) = 5$.
Solution to Exercise 3. For (a), True, 3 is an element of S. For (b), False, 4 is not in S. For (c), True, both 5 and 7 are in S, so ${5, 7}$ is a subset. For (d), False, 0 is not an element of S, so ${0}$ cannot be a subset. For (e), True, the empty set is universally a subset of every set.
Solution to Exercise 4. The subsets of $T = {x, y}$ are: $\emptyset, {x}, {y}$, and ${x, y}$. There are $2^2 = 4$ total subsets.