157k views
2 votes
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 sentence.
a) ¬p b) p ∨ q
c) ¬p ∧ q d) q → p
e) ¬q → ¬p f ) ¬p → ¬q
g) p ↔ q h) ¬q ∨ (¬p ∧ q)

User JotaBe
by
8.4k points

1 Answer

3 votes

Final answer:

The question involves translating logical propositions into English sentences. Propositions p and q represent "The election is decided" and "The votes have been counted," respectively. Compound propositions are then expressed in English using logical connectors such as not (¬), or (∨), and (∧), implying various scenarios regarding the status of the election and the vote count.

Step-by-step explanation:

In the context of propositional logic, let p represent the proposition "The election is decided," and let q represent "The votes have been counted." We can express the given compound propositions in English as follows:

a) ¬p: The election is not decided.b) p ∨ q: The election is decided or the votes have been counted (or both).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 but the votes have been counted.

These translations provide a clear understanding of how logical connectors are applied in compound propositions to convey complex relationships between individual statements.

User Etoropov
by
8.0k points