Element of a Set — Definition, Symbol & Examples
Element of a Set
A number, letter, point, line, or any other object contained in a set. For example, the elements of the set {a, b, c} are the letters a, b, and c.
Key Formula
x∈S
Where:
- x = An object that belongs to the set
- S = The set that contains the object
- ∈ = The membership symbol, read as 'is an element of' or 'belongs to'
Worked Example
Problem: Let A = {2, 4, 6, 8, 10}. Determine whether 4 and 5 are elements of set A.
Step 1: List the elements of set A. The set contains exactly the numbers 2, 4, 6, 8, and 10.
A={2,4,6,8,10}
Step 2: Check whether 4 appears in the set. Since 4 is listed inside the braces, it is an element of A.
4∈A
Step 3: Check whether 5 appears in the set. Since 5 is not listed inside the braces, it is not an element of A. We use the symbol ∉ to show this.
5∈/A
Answer: 4 is an element of A (written 4∈A), and 5 is not an element of A (written 5∈/A).
Another Example
Problem: Let B = {x | x is a positive even number less than 10}. Is 6 an element of B? Is 9?
Step 1: Interpret the set-builder notation. B contains all positive even numbers less than 10.
B={2,4,6,8}
Step 2: Since 6 is positive, even, and less than 10, it satisfies the rule and belongs to B.
6∈B
Step 3: The number 9 is positive and less than 10, but it is odd, not even. It fails the membership rule.
9∈/B
Answer: 6∈B and 9∈/B.
Frequently Asked Questions
What is the difference between ∈ and ⊂ (element vs. subset)?
The symbol ∈ means an individual object belongs to a set, while ⊂ (or ⊆) means an entire set is contained within another set. For example, 3∈{1,2,3} says 3 is a member, whereas {1,2}⊂{1,2,3} says the whole set {1, 2} is a subset.
Can a set be an element of another set?
Yes. Sets can contain other sets as elements. For instance, if C={1,{2,3}}, then C has exactly two elements: the number 1 and the set {2,3}. Note that 2 by itself is not an element of C — only the set {2,3} is.
Element vs. Subset
An element is a single object that belongs to a set, written x∈S. A subset is an entire collection of elements that all belong to another set, written A⊆S. A single element like 3 can be a member of {1,2,3}, but the set {3} (containing 3) is a subset of {1,2,3}. The distinction matters: 3∈{1,2,3} is true, but 3⊆{1,2,3} is not valid because 3 is not itself a set.
Why It Matters
The concept of set membership is the foundation of nearly all modern mathematics. Every definition in algebra, geometry, and calculus ultimately relies on specifying which objects belong to which sets. Understanding elements lets you work with domains of functions, solution sets of equations, and probability sample spaces — all of which are defined by their elements.
Common Mistakes
Mistake: Confusing an element with a subset — writing {3}∈{1,2,3} instead of 3∈{1,2,3}.
Correction: The braces {} create a set. The object 3 is an element of {1,2,3}, but {3} is a subset, not an element (unless the set explicitly contains {3} as a member). Use ∈ for individual objects and ⊆ for sets within sets.
Mistake: Assuming that because an element belongs to a subset, it also appears as a separate element in a nested set.
Correction: If S={1,{2,3}}, students often think 2∈S. It is not — only 1 and the set {2,3} are elements of S. Always look at what is directly listed between the outermost braces.
Related Terms
- Set — The collection that contains elements
- Subset — A set whose elements all belong to another set
- Empty Set — A set with no elements at all
- Universal Set — The set containing all elements under discussion
- Set-Builder Notation — A way to define elements by a rule
- Point — A geometric object that can be an element
- Line — A geometric object that can be an element