A graded collection of 20 worked exercises on the Cartesian product, developed with theoretical rigour and a focus on conceptual understanding. The exercises lead the reader from the construction of ordered pairs through to the study of subsets, relations, and geometric interpretations in the plane.
Exercise 1 — level ★☆☆☆☆
Let \( A = \{1,2\} \) and \( B = \{a,b\} \). Find the Cartesian product \( A \times B \).
Answer
\[ A \times B = \{(1,a),(1,b),(2,a),(2,b)\} \]
Solution
Formal definition
\[ A \times B = \{(x,y) \mid x \in A,\ y \in B\} \]
Interpretation
Each element of \(A\) is paired with every element of \(B\). The process is complete once all possible combinations have been generated.
Construction
With \(1\):
\[(1,a),(1,b)\]
With \(2\):
\[(2,a),(2,b)\]
Conclusion
The final set is the union of all pairs constructed above.
Remark
Order matters: \((1,a)\neq(a,1)\).
Exercise 2 — level ★☆☆☆☆
Let \( A = \{0,1\} \) and \( B = \{2,3,4\} \). Find \( A \times B \) and its cardinality.
Answer
\[ A \times B = \{(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)\} \]
\[ |A \times B| = 6 \]
Solution
Structure of the problem
Each element of \(A\) generates a “block” of pairs combined with all the elements of \(B\).
Construction
With \(0\):
\[(0,2),(0,3),(0,4)\]
With \(1\):
\[(1,2),(1,3),(1,4)\]
Cardinality
\[ |A \times B| = |A|\cdot|B| = 2\cdot3 = 6 \]
Interpretation
The Cartesian product produces a “grid-like” structure: each choice of the first coordinate is independent of the second.
Exercise 3 — level ★★☆☆☆
Let \( A = \{-1,1\} \) and \( B = \{0,2\} \). Find \( A \times B \) and interpret it in the Cartesian plane.
Answer
\[ A \times B = \{(-1,0),(-1,2),(1,0),(1,2)\} \]
Solution
Construction
With \(-1\):
\[(-1,0),(-1,2)\]
With \(1\):
\[(1,0),(1,2)\]
Geometric interpretation
The pairs correspond to points in the plane. Together they form the vertices of a rectangle.
Key observation
\[ A \times B \neq B \times A \]
Swapping the order of the sets yields different points.
Exercise 4 — level ★★☆☆☆
Let \( A = \{1,2,3\} \) and \( B = \{x\} \). Find \( A \times B \).
Answer
\[ A \times B = \{(1,x),(2,x),(3,x)\} \]
Solution
Analysis
The set \(B\) contains a single element, which fixes the second coordinate.
Construction
\[ (1,x),(2,x),(3,x) \]
Interpretation
Every pair shares the same second coordinate.
Cardinality
\[ |A \times B| = 3 \]
Exercise 5 — level ★★☆☆☆
Let \( A = \{a,b\} \) and \( B = \varnothing \). Find \( A \times B \).
Answer
\[ A \times B = \varnothing \]
Solution
Definition
Building a pair requires an element \(y \in B\).
Observation
Since \(B\) is empty, no such choice is available.
Conclusion
No pair can be formed:
\[ A \times B = \varnothing \]
General property
\[ A \times \varnothing = \varnothing \]
Exercise 6 — level ★★☆☆☆
Let \( A = \{1,2,3\} \) and \( B = \{a,b\} \). Find the subset of \( A \times B \) defined by:
\[ S = \{(x,y) \in A \times B \mid x > 1\} \]
Answer
\[ S = \{(2,a),(2,b),(3,a),(3,b)\} \]
Solution
Reading the condition
The condition \(x > 1\) selects only certain elements of \(A\).
Selection
\[ A = \{1,2,3\} \Rightarrow x > 1 \Rightarrow x \in \{2,3\} \]
Construction
With \(2\):
\[(2,a),(2,b)\]
With \(3\):
\[(3,a),(3,b)\]
Interpretation
The constraint acts only on the first coordinate, so entire “columns” are selected.
Exercise 7 — level ★★☆☆☆
Let \( A = \{1,2,3\} \) and \( B = \{1,2\} \). Find:
\[ S = \{(x,y) \in A \times B \mid x = y\} \]
Answer
\[ S = \{(1,1),(2,2)\} \]
Solution
Meaning of the condition
The relation \(x = y\) requires the two coordinates to coincide.
Element-by-element check
Possible pairs:
\((1,1)\) ✔
\((1,2)\) ✘
\((2,1)\) ✘
\((2,2)\) ✔
\((3,1)\) ✘
\((3,2)\) ✘
Conclusion
\[ S = \{(1,1),(2,2)\} \]
Remark
The pair \((3,3)\) does not appear because \(3 \notin B\).
Exercise 8 — level ★★★☆☆
Let \( A = \{1,2,3\} \). Find:
\[ A \times A \]
Answer
\[ A \times A = \{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\} \]
Solution
Structure
Here we are computing the product of a set with itself.
Construction
With \(1\):
\[(1,1),(1,2),(1,3)\]
With \(2\):
\[(2,1),(2,2),(2,3)\]
With \(3\):
\[(3,1),(3,2),(3,3)\]
Cardinality
\[ |A \times A| = |A|^2 = 3^2 = 9 \]
Interpretation
The result is a square grid: each element is also paired with itself.
Exercise 9 — level ★★★☆☆
Let \( A = \{1,2,3\} \) and \( B = \{1,2,3\} \). Find:
\[ S = \{(x,y) \in A \times B \mid x < y\} \]
Answer
\[ S = \{(1,2),(1,3),(2,3)\} \]
Solution
Meaning of the condition
The relation \(x < y\) keeps only those pairs whose first coordinate is strictly smaller than the second.
Systematic analysis
Checking each candidate:
\((1,2)\) ✔
\((1,3)\) ✔
\((2,3)\) ✔
all other pairs ✘
Geometric interpretation
The selected points lie strictly above the diagonal \(x=y\).
Exercise 10 — level ★★★☆☆
Let \( A = \{1,2\} \), \( B = \{a,b\} \), \( C = \{0,1\} \). Find:
\[ A \times B \times C \]
Answer
\[ \begin{aligned} A \times B \times C = \{ & (1,a,0),(1,a,1),(1,b,0),(1,b,1), \\ & (2,a,0),(2,a,1),(2,b,0),(2,b,1) \} \end{aligned} \]
Solution
Definition
\[ A \times B \times C = \{(x,y,z) \mid x \in A,\ y \in B,\ z \in C\} \]
Strategy
We first build \(A \times B\), then attach the third coordinate.
Construction
Each pair in \(A \times B\) yields two triples (with 0 and with 1).
Cardinality
\[ |A \times B \times C| = 2 \cdot 2 \cdot 2 = 8 \]
Interpretation
This is a Cartesian product with three factors: each element is an ordered triple.
Exercise 11 — level ★★★☆☆
Let \( A = \{1,2,3\} \) and \( B = \{1,2,3\} \). Find:
\[ S = \{(x,y) \in A \times B \mid x \ge y\} \]
Answer
\[ S = \{(1,1),(2,1),(2,2),(3,1),(3,2),(3,3)\} \]
Solution
Reading the condition
The relation \(x \ge y\) keeps every pair whose first coordinate is greater than or equal to the second.
Systematic analysis
\((1,1)\) ✔
\((2,1)\),\((2,2)\) ✔
\((3,1)\),\((3,2)\),\((3,3)\) ✔
Geometric interpretation
We obtain the part of the plane on or below the diagonal.
Exercise 12 — level ★★★☆☆
Let \( A = \{1,2,3,4\} \) and \( B = \{1,2,3\} \). Find:
\[ S = \{(x,y) \in A \times B \mid x + y = 4\} \]
Answer
\[ S = \{(1,3),(2,2),(3,1)\} \]
Solution
Meaning of the condition
The relation imposes a constraint linking the two coordinates: their sum must equal 4.
Verification
\((1,3)\) ✔
\((2,2)\) ✔
\((3,1)\) ✔
all other pairs ✘
Geometric interpretation
The selected points lie on a discrete line: \(x + y = 4\).
Exercise 13 — level ★★★★☆
Let \( A = \{1,2,3\} \). Find:
\[ S = \{(x,y) \in A \times A \mid x \neq y\} \]
Answer
\[ S = \{(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)\} \]
Solution
Interpretation
The condition removes every pair with equal coordinates.
Construction
Starting from \(A \times A\) (which has 9 elements), we discard:
\[ (1,1),(2,2),(3,3) \]
Conclusion
Six pairs remain.
Remark
\[ |S| = |A|^2 - |A| = 3^2 - 3 = 6 \]
Sets of this kind play a central role in the theory of relations.
Exercise 14 — level ★★★★☆
Let \( A = \mathbb{N} \), \( B = \mathbb{N} \). Find:
\[ S = \{(x,y) \in \mathbb{N} \times \mathbb{N} \mid y = 2x\} \]
Answer
\[ S = \{(x,2x) \mid x \in \mathbb{N}\} \]
Solution
Analysis
The set is infinite: it consists of all pairs that satisfy \(y = 2x\).
Construction
For every \(x \in \mathbb{N}\), there is a unique \(y = 2x\).
Interpretation
The set traces out a discrete line in the Cartesian plane.
Remark
It is not the whole of \( \mathbb{N} \times \mathbb{N} \), but only a single “line” inside it.
Exercise 15 — level ★★★★☆
Find the subset of \( \mathbb{R}^2 \) defined by:
\[ S = \{(x,y) \in \mathbb{R}^2 \mid y = x^2\} \]
Answer
\[ S = \text{the set of points on the parabola } y = x^2 \]
Solution
Interpretation
The set contains every real pair satisfying the relation \(y = x^2\).
Structure
It is not a discrete set, but a continuous one.
Geometric meaning
It represents a parabola in the Cartesian plane.
Key observation
The Cartesian product \( \mathbb{R}^2 \) is the entire plane, whereas \(S\) is merely a curve sitting inside it.
Exercise 16 — level ★★★★☆
Let \( A = \{1,2,3\} \). Find:
\[ S = \{(x,y) \in A \times A \mid x + y \text{ is even}\} \]
Answer
\[ S = \{(1,1),(1,3),(2,2),(3,1),(3,3)\} \]
Solution
Reading the condition
A sum is even precisely when:
- even + even
- odd + odd
Classification
\(1,3\) are odd — \(2\) is even.
Construction
\[ (1,1),(1,3),(3,1),(3,3),(2,2) \]
Interpretation
We obtain a regular structure (chessboard-like), which is fundamental in the study of relations.
Exercise 17 — level ★★★★★
Let \( A = \{1,2,3\} \). Decide whether the relation
\[ R = \{(x,y) \in A \times A \mid x \le y\} \]
is reflexive, symmetric, transitive.
Answer
Reflexive ✔ — Symmetric ✘ — Transitive ✔
Solution
Reflexivity
\[ (1,1),(2,2),(3,3) \in R \]
✔ property satisfied
Symmetry
Since \((1,2) \in R\), symmetry would require \((2,1)\) to be in \(R\) as well, but:
\[ 2 \le 1 \text{ is false} \]
✘ not symmetric
Transitivity
If \(x \le y\) and \(y \le z\), then \(x \le z\).
✔ property satisfied
Interpretation
This is the natural order relation.
Exercise 18 — level ★★★★★
Find the subset of \( \mathbb{R}^2 \) defined by:
\[ S = \{(x,y) \in \mathbb{R}^2 \mid xy = 1\} \]
Answer
\[ S = \text{the hyperbola } xy = 1 \]
Solution
Analysis
The relation links the two variables in a non-linear way.
Construction
\[ y = \frac{1}{x}, \quad x \neq 0 \]
Geometric interpretation
The result is a hyperbola consisting of two branches.
Remark
The Cartesian product covers the entire plane, but this relation singles out a single curve within it.
Exercise 19 — level ★★★★★
Let \( A = \{1,2,3\} \). Find:
\[ S = \{(x,y) \in A \times A \mid |x - y| = 1\} \]
Answer
\[ S = \{(1,2),(2,1),(2,3),(3,2)\} \]
Solution
Interpretation
The condition selects pairs at distance 1.
Construction
\((1,2)\),\((2,1)\)
\((2,3)\),\((3,2)\)
Remark
The relation is symmetric.
Graphical interpretation
The result is two diagonals running parallel to the main one.
Exercise 20 — level ★★★★★
Find the subset of \( \mathbb{R}^2 \) defined by:
\[ S = \{(x,y) \in \mathbb{R}^2 \mid y \ge x^2\} \]
Answer
\[ S = \text{the region above the parabola } y = x^2 \text{, parabola included} \]
Solution
Interpretation
This relation does not single out just a curve, but a whole region of the plane.
Structure
\[ y \ge x^2 \]
encompasses every point lying above the parabola, together with the points of the parabola itself.
Geometric meaning
We obtain an unbounded, connected region.
Closing remark
This example shows that a subset of \( \mathbb{R}^2 \) can equally well be:
- discrete
- a curve
- a region