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Logic, the foundational underpinning of mathematics, is a structured method for reasoning that allows mathematicians to deduce new truths from established principles. In a mathematical context, logic is used to rigorously define concepts and deduce the implications of assumed axioms and previously proven statements. A basic unit of thought in this context is the proposition, which is simply a declarative statement that is either true or false.

For example, the proposition '2 is greater than 1' can clearly be seen as true. Logical reasoning within mathematics often involves creating complex deductions from these fundamental building blocks. Establishing the truth value of propositions and understanding how they combine using logical connectors is an essential skill for students and professionals alike.

The glue that holds compound statements together in logic and mathematics is known as logical connectors or operators. These connectors include 'and' (conjunction), 'or' (disjunction), 'not' (negation), 'if...then...' (implication), and 'if and only if' (biconditional). Each connector has a specific definition and purpose:

• And (\textbf{\(\text{\and}\)}): A connector indicating that both linked propositions must be true for the compound statement to be true.
• Or (\textbf{\(\text{\or}\)}): A connector expressing that at least one of the propositions must be true.
• Not (\textbf{\(\text{eg}\)}): An operator signifying the opposite truth value of a proposition.
• If...then... (\textbf{\(\text{\rightarrow}\)}): Suggests a conditional relation where if the first proposition is true, then the second one must also be true.
• If and only if (\textbf{\(\text{\leftrightarrow}\)}): Implies a strong equivalence between two propositions such that both are either true or false together.

Understanding logical connectors is critical because they determine how truth values are derived from compound statements.

At the core of logic are logical propositions, which are declarative statements that hold a truth value, being either true or false. Logical propositions are distinct from opinions or questions; they are clear statements that assert a fact about the world or a mathematical system.

For instance, '7 is a prime number' is a logical proposition because it makes a clear factual claim that can be definitively evaluated as true. Logical propositions serve as the building blocks for more complex logical structures, such as compound statements, where multiple propositions are combined using logical connectors to express richer and more complex scenarios. It's essential for students to differentiate between simple propositions and compound propositions to effectively analyze an argument or proof within the realm of logical and mathematical reasoning.