Given two sets and , their symmetric difference is
Informally, contains the elements that are in exactly one of or .
(a) Let and . Compute . Draw a Venn diagram to illustrate it.
(b) Is commutative? That is, is always? Either prove it or find a counterexample.
(c) What is ? What is ? What do your answers tell you about identity elements and inverses for ?
(d) Is associative? Test with , , first, then try to argue in general.
(e)(Challenge.) What do (b)–(d) together say about the algebraic structure of on the collection of all subsets of a fixed set? Does anything surprise you?
Variant 2 — Union, Intersection, and Their Interaction
Let , , be sets. Recall: and .
(a) Let , , . Compute and . Are they equal?
(b) Prove that holds for all sets , , . (Hint: show an element belongs to one side if and only if it belongs to the other.)
(c) Does distribute over in the same way? That is, is always true? Check with the sets from (a), then prove it or find a counterexample.
(d) Show that and for all sets and . These are called the absorption laws.
(e)(Reflection.) Compare the relationship between and with that between and on . In what ways do and resemble or differ from and ? Think about commutativity, associativity, and identity elements.
Variant 3 — Arithmetic Mean as an Operation
Define an operation on by
(a) Compute and . Is commutative?
(b) Compute and . Is associative?
(c) Does have an identity element? That is, is there some with for all ?
(d) If an identity exists, does every element have an inverse under ? (An inverse of is some with .)
(e)(Reflection.) Give an intuitive explanation for why the operation has the properties you discovered. Can you think of a real situation where this “failure” causes problems?
Variant 4 — A Twisted Addition on
Define an operation on by
(a) Compute and . Is commutative?
(b) Compute and . Is associative?
(c) Find an identity element for : a vector such that for every .
(d) Given , find its inverse under : a vector such that .
(e)(Reflection.) Ordinary addition on and the operation share some properties but differ in others. Identify precisely what is the same and what is different. Does behave like in the ways that matter?
Variant 5 — A Multiplicative Twist
Define an operation on by
(a) Compute , , and .
(b) Is commutative? Associative? Test numerically first, then try to prove.
(c) Find the identity element for (if it exists).
(d) For which values of does an inverse under exist?
(e)(Challenge.) Make the substitution and , and compute in terms of and . What familiar operation does “secretly” correspond to? What does this tell you about the structure of ?
Variant 6 — Cartesian Products and Operations
Let and be sets. Recall that .
(a) Write out all elements of . How many are there?
(b) Is in general? Give a concrete example to support your answer.
(c) Compute the number of elements in and in by listing them. Based on (a) and these examples, conjecture a formula for the number of elements in in terms of the number of elements in and in .
(d) An element of is an ordered pair: the two entries play different roles. In , the points and are distinct. For which real numbers and is ?
(e) The set is often written , but this hides an ambiguity. The set has elements of the form , while has elements of the form . Are these the same set? Write down a specific element of each. Describe a natural way to match every element of one with exactly one element of the other.
Variant 7 — The Centroid of a Triangle
Let , , be three points in , thought of as position vectors. The centroid of triangle is
Recall from high school: a line through in direction is given by , , . In vector notation this is with ; the line through two points and uses direction .
(a) Compute the centroid of the triangle with vertices , , . Mark it on a sketch.
(b) A median of a triangle is the segment from a vertex to the midpoint of the opposite side. Find the midpoints of the three sides and write parametric equations for each of the three medians.
(c) Show that lies on each median.
(d) What fraction of the way from each vertex to the opposite midpoint does lie? Is this the same for all three medians?
(e)(Generalization.) Suppose you have four points in forming a tetrahedron. Conjecture a formula for the centroid. What do you think the analogous “median” property would be?
Variant 8 — Lines through Two Points
Let and be two distinct points in .
(a) Explain why the set of points of the form , , describes the line through and . This way to describe a line is called its parametric equation. What does give? ? ?
(b) Find the parametric equation of the line through and .
(c) Determine whether the point lies on this line. What about ?
(d) Three points are collinear if they all lie on a single line. Show that , , are collinear if and only if for some . Test whether , , are collinear.
(e) A classmate writes the line through and as , . Is this the same line? Is it the same parametric description? Explain.
Variant 9 — Diagonals of a Parallelogram
Let , , , be four points in forming a parallelogram (a quadrilateral with parallel opposite sides), with vertices listed in order so that and .
(a) The condition ” is a parallelogram” can be expressed as a single vector condition. Write it down and explain why it captures the geometric meaning.
(b) The diagonals of are the segments and . Find the midpoint of each diagonal, in terms of .
(c) Show that the two midpoints are equal. Conclude that the diagonals of a parallelogram bisect each other.
(d) Take , , and find so that is a parallelogram. Verify your result from (c) with these coordinates.
(e)(Converse.) Suppose is a quadrilateral (not assumed to be a parallelogram) whose diagonals bisect each other. Prove that must be a parallelogram.
Variant 10 — Varignon’s Theorem
Let , , , be any four points in forming a quadrilateral. Let , , , be the midpoints of sides , , , respectively.
(a) Express , , , in terms of , , , .
(b) Compute the vector and the vector . What do you notice?
(c) Compute and . What do you notice?
(d) Conclude that is always a parallelogram, regardless of the shape of . (This is Varignon’s theorem.)
(e) Test your result with , , , . Sketch both the original quadrilateral and the Varignon parallelogram.
Variant 11 — Scalar Multiplication and Ratios
Let and be two points in .
(a) Show that the point divides the segment in the ratio from . (Hint: write it as for a specific .)
(b) Find the point that divides the segment from to in the ratio from . Verify by sketching.
(c) The formula for dividing in ratio from is . Verify this is consistent with (a) and (b).
(d) The centroid of triangle is defined as , and a median connects a vertex to the midpoint of the opposite side. Find the midpoint of side . Then write in the form and determine . Does lie on the median from to ?
(e) Using the value of from (d) and the formula from (c), determine the ratio in which divides the median from to . A student claims this ratio is from the vertex — is the student correct?
(f)(Extension.) Does the ratio formula generalize to for any ? Write out the formula and explain why the proof requires no new ideas beyond what you used in .
Variant 12 — Lines and Points in
Recall from high school: a line through in direction is given by , , , . In vector notation this is with ; the line through two points and uses direction .
(a) Write a parametric equation for the line in passing through and .
(b) Does the point lie on this line? Does ?
(c) Two lines in are parallel if their direction vectors are scalar multiples of each other (and they do not intersect). Write down a line parallel to the one in (a) that passes through the origin.
(d) Two lines can also be skew in : neither parallel nor intersecting. Is the line from (a) skew to the line through and ?
(e) How many parameters do you need to describe a line in ? In ? In ? What does this say about the “dimension” of a line?
Variant 13 — Diagonals of a Box
A rectangular box (cuboid) has one vertex at the origin. The three edges from that vertex point in the directions of vectors , , , so the three adjacent vertices are the tips of , , .
(a) Write down all eight vertices of the box in terms of , , . (Hint: each vertex is a sum of some subset of , including the empty sum .)
(b) Take , , . List the eight vertices numerically. Make a rough sketch.
(c) A space diagonal of the box connects two vertices that share no face. There are exactly four such diagonals. Identify all four in your sketch by listing the pairs of vertices they connect.
(d) Compute the midpoint of each space diagonal using the numerical values from (b). What do you notice?
(e) Show that all four space diagonals of the box have the same midpoint, and find that midpoint in terms of , , . Compare with Variant 9 (Diagonals of a Parallelogram): what is the same, and what is new?
Variant 14 — Vectors in the Wild
A hospital records three measurements for each patient: systolic blood pressure (mmHg), resting heart rate (bpm), and blood oxygen level (%).
Patient
Blood pressure
Heart rate
Oxygen (%)
Alice
120
72
98
Bob
145
88
95
Carol
110
65
99
Denis
160
95
93
(a) Represent each patient as a vector in . What does the vector space represent in this context?
(b) Compute the “average patient” vector. What does this vector represent medically? Is it meaningful?
(c) Bob’s doctor wants to track how his measurements changed from his last visit, where his readings were . Compute the change vector. What does the sign of each component tell you?
(d) A healthy range might be (roughly): blood pressure between 90 and 130, heart rate between 60 and 100, oxygen above 95. Which patients fall within this healthy range? Justify each answer using their vectors.
(e)(Reflection.) In this problem the components of your vectors carry specific units and meanings. Does this affect whether vector operations (addition, scalar multiplication) are meaningful? When is it sensible to “add” two patient vectors, and when is it not?
Variant 15 — De Morgan’s Laws
Let be a fixed universe and let . For any set , write for the complement of in .
(a) Let , , .
Compute and . Are they equal?
Compute and . Are they equal?
(b) Prove the first De Morgan law:
(Hint: show that an arbitrary element belongs to the left-hand side if and only if it belongs to the right-hand side. At some point you will need to unpack what it means for to not belong to a union.)
(c) Prove the second De Morgan law:
(d) State and prove the analogues of De Morgan’s laws for three sets: and . (Hint: you can apply the result of (b) twice.)
(e) Use the De Morgan laws to show that union can be expressed using only intersection and complementation:
Verify this with the sets from part (a). Then write down the analogous formula expressing using only union and complementation.
Variant 16 — Vertices and Sides of a Regular Polygon
A regular hexagon is inscribed in a unit circle centred at the origin. Its six vertices, listed in order, are:
(a) Compute the sum . What do you get?
(b) Verify the same result for a square inscribed in the unit circle (vertices at and ), and for an equilateral triangle inscribed in the unit circle (vertices at , , ).
(c) For the hexagon: notice that and are opposite vertices (directly across the centre), and similarly for and . Show that each pair of opposite vertices sums to the zero vector. How does this explain the result of (a)?
(d)(Exploration.) For each edge of the hexagon, form the side vector: the displacement from one vertex to the next. That is, let for , and . Compute all six side vectors and find their sum. What do you get? Can you explain why — without computing — using only the fact that a side vector is a difference of two consecutive vertices?
(e)(Reflection.) Your explanation in (d) should work for any closed polygon with any number of sides, in any dimension. Write a brief general argument. Does your argument from (c) also generalise to any polygon? Why or why not?