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Upper Bound of a Set — Definition, Examples & Properties

Upper Bound of a Set

Any number that is greater than or equal to all of the elements of the set. For example, 5 is an upper bound of the interval [0,1]. So are 4, 3, 2, and 1.

 

 

 

See also

Least upper bound, lower bound, interval notation

Key Formula

u is an upper bound of S  ⟺  x≤u for all x∈Su \text{ is an upper bound of } S \iff x \leq u \text{ for all } x \in S
Where:
  • SS = A set of real numbers
  • uu = A real number that is an upper bound of S
  • xx = Any element of the set S

Worked Example

Problem: Determine whether 10, 7, and 6 are upper bounds of the set S = {2, 4, 6}.
Step 1: Identify the largest element of S. The elements are 2, 4, and 6, so the largest is 6.
max⁡(S)=6\max(S) = 6
Step 2: Check whether 10 is greater than or equal to every element. Since 2 ≤ 10, 4 ≤ 10, and 6 ≤ 10, the number 10 is an upper bound of S.
x≤10 for all x∈{2,4,6}✓x \leq 10 \text{ for all } x \in \{2, 4, 6\} \quad \checkmark
Step 3: Check whether 7 is an upper bound. Since 2 ≤ 7, 4 ≤ 7, and 6 ≤ 7, the number 7 is also an upper bound.
x≤7 for all x∈{2,4,6}✓x \leq 7 \text{ for all } x \in \{2, 4, 6\} \quad \checkmark
Step 4: Check whether 6 is an upper bound. Since 2 ≤ 6, 4 ≤ 6, and 6 ≤ 6, the number 6 qualifies as well. Notice that an upper bound can be equal to an element of the set.
x≤6 for all x∈{2,4,6}✓x \leq 6 \text{ for all } x \in \{2, 4, 6\} \quad \checkmark
Answer: All three numbers — 10, 7, and 6 — are upper bounds of {2, 4, 6}. The smallest upper bound (least upper bound) is 6.

Another Example

Problem: Is 0.5 an upper bound of the open interval (0, 1)?
Step 1: The set (0, 1) contains all real numbers strictly between 0 and 1, such as 0.3, 0.7, and 0.999.
Step 2: For 0.5 to be an upper bound, every element of (0, 1) must satisfy x ≤ 0.5. But 0.8 is in (0, 1) and 0.8 > 0.5.
0.8∈(0,1) and 0.8>0.50.8 \in (0,1) \text{ and } 0.8 > 0.5
Step 3: Since at least one element exceeds 0.5, the number 0.5 is not an upper bound of (0, 1). Any number ≥ 1 is an upper bound, and the least upper bound is 1 — even though 1 is not in the set.
Answer: No, 0.5 is not an upper bound of (0, 1). The least upper bound is 1.

Frequently Asked Questions

Does an upper bound have to be in the set?
No. An upper bound can lie outside the set entirely. For example, 5 is an upper bound of {1, 2, 3} even though 5 is not an element of that set. If the set has a maximum element, that maximum is also an upper bound (and in fact the least upper bound), but many upper bounds will lie outside the set.
Can a set have more than one upper bound?
Yes — if a set has any upper bound at all, it has infinitely many. If u is an upper bound of S, then every number greater than u is also an upper bound of S. Among all upper bounds, the smallest one is called the least upper bound or supremum.

Upper Bound vs. Lower Bound

An upper bound of a set S is any number greater than or equal to every element of S. A lower bound is any number less than or equal to every element of S. For instance, for the set [2, 8], any number ≥ 8 is an upper bound and any number ≤ 2 is a lower bound. A set that has both an upper and a lower bound is called a bounded set.

Why It Matters

Upper bounds are fundamental in calculus and real analysis. The Least Upper Bound Property (completeness) of the real numbers — which states that every nonempty set with an upper bound has a least upper bound — is what distinguishes the reals from the rationals and underpins the rigorous definitions of limits, integrals, and continuity. In optimization, identifying upper bounds helps you determine the maximum value a function or quantity can achieve.

Common Mistakes

Mistake: Thinking the upper bound must be the largest element of the set.
Correction: Any number at or above every element counts as an upper bound, and a set typically has infinitely many upper bounds. The special one that equals the largest value (or approaches it) is the least upper bound, which is a distinct concept.
Mistake: Assuming the upper bound must belong to the set.
Correction: An upper bound does not need to be an element of the set. For the open interval (0, 1), the number 1 is the least upper bound even though 1 ∉ (0, 1).

Related Terms