How to Work Out Upper and Lower Bounds
Understand the concept of an upper bound., Understand the concept of a lower bound., Check to see if your set is bounded from above., Check to see if your set is bounded from below., Determine whether your set has a supremum., Determine whether your...
Step-by-Step Guide
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Step 1: Understand the concept of an upper bound.
If a set of real numbers, labeled S, includes a real number A ∈ R such that every number of the subset S is less than or equal to A, then S is said to be “bounded from above.” A is an upper bound.
Mathematically, this is expressed as follows: ∀x∈S⇒x≤A.
If S does not have an upper bound, it is said to be “unbounded from above.” If there is a least member among the upper bounds of the set S, then this number is called the “least upper bound” or the “supremum” of the set and is denoted by supS.
If a set S has at least one upper bound, then there are infinitely many upper bounds greater than that number. -
Step 2: Understand the concept of a lower bound.
If a set of real numbers, labeled S, includes a real number B ∈ R such that every number of the subset S is great than or equal to B, then S is said to be “bounded from below.” B is a lower bound.
Mathematically, this is expressed as follows: ∀x∈S ⇒x≥B If S does not have a lower bound, it is said to be “unbounded from below.” If there is a greatest member among the lower bounds of the set S, then this member is called the “greatest lower bound” or “infimum” of the set and is denoted by infS.
If a set S has at least one lower bound, then there are infinitely many lower bounds less than that number. , If for a set of real numbers, S, ∃A∈R such that ∀x∈S ⇒x≤A, then A is said to be an upper bound of S.
In other words, if there is a real number A so that any number picked from the set of numbers is less than or equal to it, the set is indeed bounded from above.
For example, say you have the follow set of real numbers, S: {1,
-1/4, 1/9, 1/16 . . .}.
In this example, there is a real number A that is equal to 1, and any number from the set will be less than or equal to it.
Therefore, the set is bounded from above. , If for a set of real numbers, S, ∃B∈R such that ∀x∈S⇒x≥B, then B is said to be a lower bound of S.
In other words, if there is a real number B so that any number picked from the set of numbers is greater than or equal to it, the set is indeed bounded from below.
In the example above, there is a real number B that is equal to
-1/4, and any number from the set will be greater than or equal to it.
Therefore, the set is bounded from below. , If there is a lowest number among the upper bounds of the set, then this number is the supremum, denoted supS.
In the example above, any number greater than 1 would be an upper bound, but 1 is the lowest upper bound.
Therefore, 1 is your supremum: supS =
1. , If there is a greatest number among the lower bounds of the set, then this number is the infimum, denoted infS.
In the example above, any number less than
-1/4 would be a lower bound, but
-1/4 is the greatest lower bound.
Therefore,
-1/4 is your infimum: infS =
-1/4. , A number a is the greatest member of a set S if a∈S⋀x∈S⇒x≤a.
In other words, if you pick a number from the set, and any number compared to that number is less than or equal to it, that number is the greatest member of the set.
It is also called the “maximum.” In the example above, there is indeed a number a such that these conditions exist.
That number is 1, and 1 is therefore the greatest member of your set. , A number b is the least member of a set S if b∈S⋀x∈S⇒x≥b.
In other words, if you pick a number from the set, and any number compared to that number is greater than or equal to it, that number is the lowest member of the set.
It is also called the “minimum.” In the example above, there is indeed a number b such that these conditions exist.
That number is
-1/4, and
-1/4 is therefore the lowest member of your set. , The biggest and smallest numbers in your set are the upper and lower bounds.
In the example above, you have a set that is both bounded from above and bounded from below, by 1 and
-1/4. -
Step 3: Check to see if your set is bounded from above.
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Step 4: Check to see if your set is bounded from below.
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Step 5: Determine whether your set has a supremum.
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Step 6: Determine whether your set has an infimum.
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Step 7: Find the greatest member of your set.
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Step 8: Find the lowest member of your set.
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Step 9: Note the upper and lower bounds of your set.
Detailed Guide
If a set of real numbers, labeled S, includes a real number A ∈ R such that every number of the subset S is less than or equal to A, then S is said to be “bounded from above.” A is an upper bound.
Mathematically, this is expressed as follows: ∀x∈S⇒x≤A.
If S does not have an upper bound, it is said to be “unbounded from above.” If there is a least member among the upper bounds of the set S, then this number is called the “least upper bound” or the “supremum” of the set and is denoted by supS.
If a set S has at least one upper bound, then there are infinitely many upper bounds greater than that number.
If a set of real numbers, labeled S, includes a real number B ∈ R such that every number of the subset S is great than or equal to B, then S is said to be “bounded from below.” B is a lower bound.
Mathematically, this is expressed as follows: ∀x∈S ⇒x≥B If S does not have a lower bound, it is said to be “unbounded from below.” If there is a greatest member among the lower bounds of the set S, then this member is called the “greatest lower bound” or “infimum” of the set and is denoted by infS.
If a set S has at least one lower bound, then there are infinitely many lower bounds less than that number. , If for a set of real numbers, S, ∃A∈R such that ∀x∈S ⇒x≤A, then A is said to be an upper bound of S.
In other words, if there is a real number A so that any number picked from the set of numbers is less than or equal to it, the set is indeed bounded from above.
For example, say you have the follow set of real numbers, S: {1,
-1/4, 1/9, 1/16 . . .}.
In this example, there is a real number A that is equal to 1, and any number from the set will be less than or equal to it.
Therefore, the set is bounded from above. , If for a set of real numbers, S, ∃B∈R such that ∀x∈S⇒x≥B, then B is said to be a lower bound of S.
In other words, if there is a real number B so that any number picked from the set of numbers is greater than or equal to it, the set is indeed bounded from below.
In the example above, there is a real number B that is equal to
-1/4, and any number from the set will be greater than or equal to it.
Therefore, the set is bounded from below. , If there is a lowest number among the upper bounds of the set, then this number is the supremum, denoted supS.
In the example above, any number greater than 1 would be an upper bound, but 1 is the lowest upper bound.
Therefore, 1 is your supremum: supS =
1. , If there is a greatest number among the lower bounds of the set, then this number is the infimum, denoted infS.
In the example above, any number less than
-1/4 would be a lower bound, but
-1/4 is the greatest lower bound.
Therefore,
-1/4 is your infimum: infS =
-1/4. , A number a is the greatest member of a set S if a∈S⋀x∈S⇒x≤a.
In other words, if you pick a number from the set, and any number compared to that number is less than or equal to it, that number is the greatest member of the set.
It is also called the “maximum.” In the example above, there is indeed a number a such that these conditions exist.
That number is 1, and 1 is therefore the greatest member of your set. , A number b is the least member of a set S if b∈S⋀x∈S⇒x≥b.
In other words, if you pick a number from the set, and any number compared to that number is greater than or equal to it, that number is the lowest member of the set.
It is also called the “minimum.” In the example above, there is indeed a number b such that these conditions exist.
That number is
-1/4, and
-1/4 is therefore the lowest member of your set. , The biggest and smallest numbers in your set are the upper and lower bounds.
In the example above, you have a set that is both bounded from above and bounded from below, by 1 and
-1/4.
About the Author
Jeffrey Ellis
Brings years of experience writing about pet care and related subjects.
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