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)

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.

Explore compound propositions in propositional logic with detailed English translations. Learn about negation, disjunction, conjunction, implication, biconditional statements, and their applications. Suitable for advanced high school or college-level students.

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