Math Problem Statement

Let p and q be the propositions “The election is decided” and “The votes have been counted,” respectively. Express

each of these compound propositions as an English sen- tence.

a) ¬p b) p ∨ q c) ¬p ∧ q d) q → p e) ¬q → ¬p f ) ¬p → ¬q g) p ↔ q h) ¬q ∨ (¬p ∧ q)

Solution

Here are the compound propositions expressed as English sentences:

a) ¬p: "The election is not decided."

b) p ∨ q: "The election is decided, or the votes have been counted."

c) ¬p ∧ q: "The election is not decided, and the votes have been counted."

d) q → p: "If the votes have been counted, then the election is decided."

e) ¬q → ¬p: "If the votes have not been counted, then the election is not decided."

f) ¬p → ¬q: "If the election is not decided, then the votes have not been counted."

g) p ↔ q: "The election is decided if and only if the votes have been counted."

h) ¬q ∨ (¬p ∧ q): "Either the votes have not been counted, or the election is not decided and the votes have been counted."

Would you like more details or have any questions?

Here are 5 related questions to explore:

  1. What does the logical negation (¬) of a proposition mean?
  2. How does the logical operator "or" (∨) differ from "and" (∧) in compound propositions?
  3. What is the significance of "if and only if" (↔) in logical statements?
  4. Can a proposition be both true and false simultaneously? Why or why not?
  5. How does the implication (→) work in logical statements?

Tip: When translating logical propositions into English, carefully consider the meaning of each logical operator (¬, ∨, ∧, →, ↔) to ensure the correct interpretation.