Answer to Question #310311 in Discrete Mathematics for Ali

Question

Let p = “The Exams are decided” and q = “The Papers have been set”

Express each of these compound propositions as English sentences. And state

whether it is Contradiction, Tautology or Contingency.

a) ¬ p :

b) p ∨ q :

c) q → p :

d) p ↔ q :

e) ¬ p ∧ q :

f) ¬ p → ¬ q :

g) ¬ q → ¬ p :

h) ¬ q ∨ (¬ p ∧ q) :

Answer 1

Given that

$p =$ “The Exams are decided” and $q =$ “The Papers have been set”

Solution (a)

$¬ p=$ ” The exams are not decided”

Solution (b)

$p ∨ q =$ “The exams are decided, or the papers have been set”

Solution (c)

$q → p =$ “If the exams have been decided then the papers have been set”.

Solution (d)

$p ⟷ q =$ “The exams are decided if and only if the papers have been set”

Solution (e)

$¬ p ∧ q =$ “The exams are not decided, and the papers have been set”

Solution (f)

$¬ p → ¬ q =$ ” If the exams are not decided, then the papers have not been set”

Solution (g)

$¬ q → ¬ p =$ ” If the paper has not been set then the exams are not decided”

Solution (h)

$¬ q ∨ (¬ p ∧ q)=$ "Either the papers have not been set or the papers have been set but the exams are not decided"