MAT188H1 Chap.3 Sets and Geometric Loci
Sets and Geometric Loci
A set is determined by its members
A set is a collection of objects called elements. Membership notation answers whether a particular object belongs to that collection. For a finite set, listing its elements inside braces is often convenient. Changing the order of that list does not change the set. This differs from a coordinate vector, where changing entry order usually changes the vector.
Keep braces for a collection distinct from the ordered coordinates of one point.
The empty set contains no elements. It can arise naturally when a system of membership conditions cannot be satisfied. An empty collection is different from a collection containing the number zero: the latter has an element.
Before simplifying any notation, identify whether the expression names an individual object or a set whose members must be tested.
State both the rule and the allowed inputs
Set builder notation describes elements by a property or by a rule that generates them. The domain matters as much as the formula.
A parameter ranging over the integers can produce a different set from the same expression with a parameter ranging over all real numbers. Read the vertical bar as “such that” and translate the complete statement into a sentence before using it.
For geometric sets, specify the surrounding coordinate space and the condition a point must satisfy.
A planar equation such as y = 4x - 3 defines all ordered pairs obeying that equation. Testing membership means substituting the coordinates and checking equality.
Merely having two real coordinates establishes membership in the surrounding plane, but it does not establish membership in the particular line.
Containment and equality have different proof burdens
A subset claim says that every element of the first set also belongs to the second. You can disprove it with a single counterexample, but checking a few successful elements does not prove a universal claim.
To establish containment, start with an arbitrary member of the first set and use its defining property to show that it satisfies the second set's condition.
Equality requires containment in both directions. This is especially useful when the same geometric object has different descriptions. First show that every point generated by a proposed parameter rule satisfies the original equation.
Then begin with an arbitrary point satisfying the equation and identify a parameter value that generates it. The second direction ensures the rule does not miss part of the original set.
Union permits either condition; intersection requires both
The union of two sets contains objects belonging to either set, including objects belonging to both. The intersection contains only the common elements.
These logical words determine which membership test to perform. For a union, one successful condition is enough. For an intersection, failure of either condition excludes the object. Do not replace “and” with “or” while translating a geometric description.
When sets are defined by equations, an intersection imposes the equations simultaneously.
A point found from one equation is only a candidate until the other equation is checked. A union instead keeps every point satisfying at least one equation. These operations generally describe different collections even when they start from the same pair of lines.
Equations and parameters can describe the same locus
The set y = 4x - 3 can also be written as all points (t, 4t - 3), with t real.
Every generated point satisfies the equation because its second coordinate is four times its first minus three. Conversely, a point (x, y) satisfying the equation is generated by choosing t = x. Both directions together establish that these are descriptions of exactly the same line.
A few plotted points help visualize that line, but the unrestricted real parameter supplies its complete description.
Restricting the parameter would need a separate justification and may change the object. For example, choosing only integer values gives a discrete collection of points on the line. Choosing all real values allows intermediate points as well.
The range of the parameter must therefore stay attached to the displayed expression.
Once points are identified with their position vectors, this set language connects directly to vector geometry. A statement about all points on a line becomes a statement about the endpoints of vectors in standard position.
The useful habit is to preserve the membership condition while changing representation, so that a diagram, an equation and a parameter expression describe the same objects.
What this chapter covers
- 01
Membership and set builder notation
- 02
Subset and equality proofs
- 03
Union, intersection and geometric solution sets
Worked example · free
Intersecting two line sets
- 2A common point must satisfy both expressions for y. Equate them: 2x + 1 = -x + 7. Adding x and subtracting 1 gives 3x = 6, so x = 2.
- 2Substitute x = 2 into either equation to obtain y = 5. Check the other equation as well: -2 + 7 = 5. Thus (2, 5) passes both membership tests.
- 2For (0, 1), the first equation gives 1 = 2(0) + 1, which is true. The second would require 1 = 7, which is false. Membership in A alone is sufficient for the union.
Key terms
- Set
- A collection determined by its elements. Listing order does not affect a set, while an equation or generation rule can describe a collection too large to list member by member.
- Subset
- A set whose every element also belongs to another specified set. Proving a subset relationship requires an argument for an arbitrary member; one excluded member is enough to refute it.
- Set Equality
- The relationship between sets containing exactly the same elements. Mutual containment provides a useful proof method when the two collections are presented using different notation or defining conditions.
- Union
- The collection of objects belonging to at least one of the sets being combined. An object shared by both sets remains included; “or” does not exclude the shared case.
- Intersection
- The collection of elements common to the sets being compared. In an equation description, every defining condition must hold for the same object before it belongs to the intersection.
- Parameter Domain
- The collection of allowed values for a parameter in a generated set. Altering this domain can change the resulting geometric object even when the coordinate expressions remain unchanged.
Sets and Geometric Loci FAQ
How can one counterexample refute a subset statement?
A containment claim applies to every member of its first set. Finding an element there that fails the second set membership test contradicts that universal requirement. Write the example explicitly and verify both parts: its inclusion in the first collection and its exclusion from the proposed containing collection.
Are repeated sample points enough to establish a parameterization?
Samples demonstrate that the rule works at those chosen inputs, but they do not show that it reaches every required point. Use an arbitrary point satisfying the original condition and solve for a parameter that produces it. This reverse step is the part that establishes complete coverage.
What changes when a parameter is restricted to integers?
Integer inputs exclude the intervening real parameter values. For a nonconstant linear coordinate rule, this selects separated points from the full real-parameter line. The restriction must be retained in the notation because identical coordinate formulas can describe different collections when their allowed input sets differ.
Exam move
Translate each set expression into ordinary language, then test one included and one excluded object. Practise equality using two separate containment arguments. For geometric examples, write a membership equation and a parameter description side by side, checking the domain and the reverse direction of the correspondence.
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