ECON6012 / ECON2125: Semester Two, 2024
Tutorial 1 Questions
A Note on Sources
These questions do not originate with me. They have either been influenced by, or directly drawn from, other sources.
Key Concepts
Methods of Proof, Proof by Deduction, Proof by Induction, Proof by Con-tradiction, Proof by Contraposition, Disproof by Counterexample, Binary Relations, Potential Properties of Binary Relations, Weak Completeness, Re-flexivity, Irreflexivity, Symmetry, Antisymmetry, Transitivity, Strong Com-pleteness, Rationality, Equivalence Relations, Partitions, Partially Ordered Sets, Ordered Sets (or Totally Ordered Sets), Well Ordered Sets, Weak Pref-erence Relations, Indifference Relations, Strict Preference Relations.
Tutorial Questions
Tutorial Question 1
Tutorial Question 2
Show that √
3 ∈/ Q, where
Q := {b/a : a ∈ Z, b ∈ N}
is the set of rational numbers,
Z := {· · · , −3, −2, −1, 0, 1, 2, 3, · · · }
is the set of integers, and
N := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, · · · }
is the set of natural numbers.
Tutorial Question 3
Either show that following claim is true or provide a counter-example that establishes that it is false.
Claim: The strict preference relation is antisymmetric if and only if the weak preference relation is strongly complete.
Tutorial Question 4
Consider the set A = {1, 2, 3}. Show that the set equality relation is an equivalence relation on the power set of A.
• Some background facts.
– Weak Subset: Let X and Y be two sets. X ⊆ Y if and only if every element of X is also an element of Y .
– Proper Subset: Let X and Y be two sets. X ⊂ Y if and only if both X ⊆ Y and Y 6⊆ X.
– Set Equality: Let X and Y be two sets. X = Y if and only if both X ⊆ Y and Y ⊆ X.
– Power Set: Let X be a set. The power set of X, which is often denoted by 2X, is the set of all subsets of X.
Additional Practice Questions
Additional Practice Question 1
Suppose that X is the consumption set for some consumer and % is a rational weak preference relation that is defined on X (or, more accurately, X × X). (Recall that a weak preference relation is rational if it either (i) weakly complete, reflexive, and transitive, or, equivalently, (ii) strongly complete and transitive.)
1. Show that the indifference relation, ∼, on X is reflexive, symmetric and transitive, but not strongly complete.
2. Use your answer to part one of this question to conclude that the indifference relation on X is an equivalence relation.
3. Suppose that x ∈ X. Define the indifference set for x as
I (x) = {y ∈ X : y ∼ x} .
Show that, for all (x, y) ∈ X × X, either I (x) = I (y) or I (x)∩I (y) = ∅.
4. Show that for every x ∈ X, we have I (x) = ∅.
5. Use your answers to part two and part three of this question to argue that the set of all indifference sets, {I(x) : x ∈ X}, partitions the ele-ments of X into a set of non-overlapping subsets of X without leaving out any elements of X.