Let p and q be the propositions "The election is decided"...

Let $p$ and $q$ be the propositions "The election is decided" and "The votes have been counted, " respectively. Express each of these compound propositions an English sentence. a) $\neg p$ b) $p \vee q$ c) $\neg p \wedge q$ d) $q \rightarrow p$ e) $\neg q \rightarrow \neg p$ f) $\neg p \rightarrow \neg q$ g) $p \leftrightarrow q$ h) $\neg q \vee(\neg p \wedge q)$

Let $p$ and $q$ be the propositions "The election is decided" and "The votes have been counted, " respectively. Express each of these compound propositions an English sentence.
a) $\neg p$
b) $p \vee q$
c) $\neg p \wedge q$
d) $q \rightarrow p$
e) $\neg q \rightarrow \neg p$
f) $\neg p \rightarrow \neg q$
g) $p \leftrightarrow q$
h) $\neg q \vee(\neg p \wedge q)$

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Key Concepts

Biconditional

The biconditional connective, often phrased as 'if and only if', indicates that two propositions are equivalent; that is, they are both true or both false, ensuring the same truth value.

Implication

Implication is a conditional logical connective often expressed in an 'if-then' format, which states that if one proposition (the antecedent) is true, then another proposition (the consequent) must also be true.

Conjunction

Conjunction is the logical operation representing 'and'. It forms compound statements from two propositions and yields a truth value that is true only when both propositions are true.

Negation

Negation is a logical operation that reverses the truth value of a proposition. When a proposition is negated, if it was originally true it becomes false, and vice versa.

Propositions

Propositions are declarative statements that can be clearly evaluated as true or false. They serve as the fundamental units in logic upon which compound statements are built.

Disjunction

Disjunction is a logical connective that represents 'or' in logical expressions. It combines two propositions and results in a truth value that is true when at least one of the propositions is true.

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