From a Domain Description to a Set
A domain is a set of allowed inputs. For a function of two variables, each input is an ordered pair \((x,y)\); for a function of three variables, each input is an ordered triple \((x,y,z)\). Set-builder notation describes such a set by naming its elements and stating the conditions those elements must satisfy.
This notation is useful because a verbal description can be converted into a precise membership test. To decide whether a point belongs to the set, check its coordinates against every stated condition. The order of the coordinates matters: \((x,y)\) and \((y,x)\) need not represent the same point or satisfy the same conditions.
The Parts of a Set-Builder Description
A clear set-builder description usually has three parts. First, it names the set, such as \(D\). Second, it states what kind of object is an element: an ordered pair from \(\mathbb{R}^2\), or an ordered triple from \(\mathbb{R}^3\). Third, it states the condition or conditions that determine membership.
This says that \(D\) consists of all ordered pairs of real numbers whose first coordinate is nonnegative and whose second coordinate is less than \(4\). Both conditions must hold. The comma here separates simultaneous conditions; it has the same role as “and.” It does not mean that a point may satisfy either condition alone.
Writing the ambient space is important. The notation \((x,y)\in\mathbb{R}^2\) makes clear that the elements are ordered pairs of real numbers. Likewise, \((x,y,z)\in\mathbb{R}^3\) identifies ordered triples. When describing a domain, use the coordinates in the same order as the inputs of the function or as the description specifies.
A condition may be an equality, an inequality, or a combination of both. For instance, “the second coordinate is twice the first” becomes \(y=2x\); “the first coordinate is positive” becomes \(x>0\). If both statements are required, write both conditions together.
Why the Conditions Must Match the Words
Translating into set-builder notation is not just a matter of putting braces around a formula. Each word that restricts membership must be represented by a condition, and the conditions must use the correct variables. A useful way to check a translation is to read it in both directions: read the original description and form the conditions, then read the conditions back as an ordinary sentence.
Decide whether the set consists of ordered pairs or ordered triples, and state the corresponding real coordinate space.
Turn each statement about a coordinate, distance, or relation into an equality or inequality involving the named variables.
Use “and” when every restriction is required. Keep the coordinate order fixed throughout.
Substitute selected points into every condition and confirm that the outcome agrees with the verbal description.
Worked Example: A Line with a Restricted First Coordinate
Worked Example: A Line with a Restricted First Coordinate
Write the set of planar points whose first coordinate is at least \(-2\) and less than \(1\), and whose second coordinate equals \(2x+1\).
The points are ordered pairs, so we begin with \((x,y)\in\mathbb{R}^2\). The first-coordinate restriction gives \(-2\leq x<1\). The required relationship between the coordinates gives \(y=2x+1\). Both restrictions must hold, so the set is
Check the translation using \((-2,-3)\). Its first coordinate satisfies \(-2\leq -2<1\). The equality condition gives \(2(-2)+1=-4+1=-3\), which is its second coordinate. Thus \((-2,-3)\in D\).
Now test \((1,3)\). The equality holds because \(2(1)+1=3\), but the coordinate restriction fails: \(1<1\) is false. Since both conditions are required, \((1,3)\notin D\). Finally, \((0,0)\) satisfies \(-2\leq 0<1\), but it fails the equality because \(2(0)+1=1\neq 0\). Therefore \((0,0)\notin D\). These checks confirm that the notation imposes both the coordinate restriction and the line equation.
Translating More Than One Geometric Condition
A verbal description often combines different kinds of conditions. For example, it may restrict a point by its distance from a fixed center and by the sign of one coordinate. Write a separate mathematical condition for each requirement, then place them together in the set-builder description.
For a point \((x,y)\), its distance from the origin is \(\sqrt{x^2+y^2}\). Thus a distance restriction can be expressed using that formula. When the distance and the bound are nonnegative, squaring both sides gives an equivalent inequality involving \(x^2+y^2\). The coordinate condition remains a separate requirement.
Worked Example: Points Near the Origin with Positive Second Coordinate
Write the set of planar points whose distance from the origin is less than \(3\) and whose second coordinate is positive.
The points are in \(\mathbb{R}^2\). The distance condition is \(\sqrt{x^2+y^2}<3\). Since the distance is nonnegative and \(3\) is positive, squaring both sides preserves the inequality and gives \(x^2+y^2<9\). The second-coordinate condition is \(y>0\). Hence
Check \((1,2)\). The squared distance is \(1^2+2^2=1+4=5\), and \(5<9\). Also, \(2>0\). Both conditions hold, so \((1,2)\in D\).
Check \((0,3)\). The squared distance is \(0^2+3^2=9\), so the condition \(9<9\) is false. Therefore \((0,3)\notin D\), regardless of its positive second coordinate. Check \((1,-1)\): its squared distance is \(1^2+(-1)^2=1+1=2<9\), but its second coordinate fails the condition \(-1>0\). Thus \((1,-1)\notin D\). The examples confirm that being close enough to the origin alone is not sufficient; the coordinate condition must also hold.
Writing Conditions for Ordered Triples
The same method applies in three variables. A point in \(\mathbb{R}^3\) is written as \((x,y,z)\), and each condition must refer to the appropriate coordinate. For example, a statement about height above the \(xy\)-plane is a condition on \(z\), because the \(xy\)-plane consists of points with \(z=0\).
Ordinary wording matters when translating that condition. “Above the \(xy\)-plane” means strictly above it, so \(z>0\). “On or above the \(xy\)-plane” includes the plane and is expressed by \(z\geq 0\). These are different sets: the origin has \(z=0\), so it does not satisfy \(z>0\).
Worked Example: Points in a Sphere and Strictly Above the \(xy\)-Plane
Write the set of points inside or on the sphere of radius \(3\) centered at the origin that also lie strictly above the \(xy\)-plane.
The points are ordered triples in \(\mathbb{R}^3\). The distance of \((x,y,z)\) from the origin is \(\sqrt{x^2+y^2+z^2}\). Being inside or on the sphere means this distance is at most \(3\). Both sides are nonnegative, so squaring gives the equivalent condition \(x^2+y^2+z^2\leq 9\). Strictly above the \(xy\)-plane means \(z>0\). Therefore the set is
Check \((1,2,2)\). The sum of the squared coordinates is \(1^2+2^2+2^2=1+4+4=9\), so it satisfies \(9\leq 9\). Its third coordinate satisfies \(2>0\). Thus \((1,2,2)\in D\).
Check the origin \((0,0,0)\). Its squared distance is \(0^2+0^2+0^2=0\), so \(0\leq 9\). However, its third coordinate does not satisfy \(0>0\). Thus \((0,0,0)\notin D\). Check \((0,0,3)\): the squared-coordinate sum is \(0+0+9=9\), and \(3>0\), so \((0,0,3)\in D\). Each check applies both conditions in the set description.
A Short Justification for the Translation Method
Why does combining conditions with “and” describe exactly the intended set? The reason follows directly from the meaning of set-builder notation. Suppose a description requires points to satisfy conditions \(C_1\) and \(C_2\). A point belongs to the described set precisely when it is an allowed kind of element and both \(C_1\) and \(C_2\) are true. That is exactly what the set-builder expression states.
This is also why leaving out one condition changes the set. In the sphere example, the inequality alone describes all points inside or on the sphere, including points below the plane. The additional condition \(z>0\) excludes those points and the points on the plane. A complete description must record every requirement in the original wording.
Common Mistake: Replacing “Above” with “On or Above”
A frequent translation error is to use a non-strict inequality when the description is strict. Consider the statement “the point lies above the \(xy\)-plane.” Writing \(z\geq 0\) is incorrect for this wording: it describes points on the plane as well as points above it.
The error can be seen by testing the origin. It satisfies \(z\geq 0\) because \(0\geq 0\), but it lies on the \(xy\)-plane, not above it. The correction is to use \(z>0\). If the intended description is “on or above the \(xy\)-plane,” then \(z\geq 0\) is appropriate. Check the wording before choosing the inequality.
Why Precise Set Descriptions Matter
Set-builder notation makes a domain explicit enough to use in later work. It identifies the input variables, fixes their order, and records the restrictions that determine whether an input is allowed. A reader can test any proposed point by substitution rather than guessing from a vague description.
Before accepting a description, check three things:
- Element type: Does the notation use ordered pairs for two coordinates and ordered triples for three?
- Coordinate roles: Does each condition use the variable corresponding to the intended coordinate?
- Completeness: Are all restrictions present, joined so that every required condition must hold?
Then test at least one point that should belong and one point that should not. A successful check does not prove every possible membership decision, but it often exposes a missing condition, a reversed coordinate, or an inequality that does not match the words.
Check Your Understanding
Write each set in set-builder notation. Then test a point that satisfies the conditions and a point that fails at least one condition.
- Planar points whose first coordinate is positive and whose second coordinate is \(4\).
- Planar points satisfying \(x^2+y^2\leq 16\) and \(x<0\).
- Points in \(\mathbb{R}^3\) whose coordinates satisfy \(x+y+z=5\) and \(z\geq 0\).
- Explain why a point with \(z=0\) does not satisfy the condition “strictly above the \(xy\)-plane.”
- In a set of ordered pairs, what does the condition \(y=2x\) say, and why does the order of the coordinates matter?