MAU 22C00 Trinity College The Cartesian Product of Natural Numbers Questions


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1) (10 points) Please carry out the following proof in propositional logic
following the proof format in tutorial 1: Hypotheses: P → (Q ↔ ¬R),
P ∨ ¬S, R → S, ¬Q → ¬R. Conclusion: ¬R.
For each line of the proof, mention which tautology you used giving its
number according to the list of tautologies posted in folder Course Documents. Solutions based on truth tables or any other method except
for the one specified will be given NO CREDIT.
2) (10 points) Prove the following statement: If n is any integer, then
n2 − 3n must be even. (Hint: Cases come in handy here. See tautology
#26 for the basis of proofs by√cases. This proof follows the format of
the one given in lecture that 2 is not a rational number.)
3) (10 points) Prove via inclusion in both directions that for any three
sets A, B, and C
A ∩ (B C) = (A ∩ B) (A ∩ C).
Venn diagrams, truth tables, or diagrams for simplifying statements in
Boolean algebra such as Veitch diagrams are NOT acceptable and will
not be awarded any credit.
4) (10 points) Let N × N be the Cartesian product of the set of natural
numbers with itself consisting of all ordered pairs (x1 , x2 ) such that
x1 ∈ N and x2 ∈ N. We define a relation on its power set P(N × N) as
follows: ∀A, B ∈ P(N × N) A ∼ B iff (A B) ∪ (B A) = C and C
is a finite set. Determine whether or not ∼ is an equivalence relation
and justify your answer by checking each of the three properties in the
definition of an equivalence relation. Please note that a set C is finite
if it has finitely many elements. In particular, the empty set ∅ has zero
elements and is thus finite.

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Propositional logic

boolean algebra


equivalence relation

the Cartesian product

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