Mathematics - 2023
Insper
PhD Program in Business Economics
Problem Set 1
1. Show that for all statements p, q, q1eq2:
(a) p⇔ ¬ (¬p).
(b) ¬(p∧q)⇔(¬p)∨(¬q).
(c) ¬(p∨q)⇔(¬p)∧(¬q).
(d) p∨(q1∧q2)⇔(p∨q1)∧(p∨q2).
(e) p∧(q1∨q2)⇔(p∧q1)∨(p∧q2).
(f) p⊻(q1⊻q2)⇔(p⊻q1)⊻q2.1
2. Given two statements pand q, is it always true that2
q⇒(p⇒q)?
3. Show that the compound propositions P in the following items are tau-
tologies:
(a) P:p⇒(p∨q)
(b) P: (p∧q)⇒p
(c) P: [(p⇒q)∧(q⇒q1)] ⇒[p⇒q1]. 3
(d) P: (p∧q)⇒(p∨q)
1We note that this property implies that given A,B, C ∈2X, where X=∅, it follows that:
(A∆B) ∆C=A∆ (B∆C).
2A statement that is always true is called a tautology. For instance, given p, the statement
[p∨ ¬p] is a tautology (note that we cannot combine pwith ¬pby taking T(F) and T(F) at
the same time):
p¬p[p∨ ¬p]
T F T
F T T
In many cases, a tautology is an expression or phrase that says the same thing twice, just
in a different way. In special, the equivalences found in the items (a) to (f) in the previous
exercise are tautologies.
On the other hand, a statement is called a contradiction if it is always false. For instance,
given p, the statement [p∧ ¬p] is a contradiction:
p¬p[p∧ ¬p]
T F F
F T F
A statement that assumes values T and F is called a contingency. For instance, given pand
q, by what we already see, [p∧q],[p∨q],[p⇒q], and [p⇔q] are examples of contingencies.
3This is called the law of syllogism or reasoning by transitivity
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