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- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/02%3A_Logic_and_Quantifiers/2.03%3A_Logical_EquivalencesSome logical statements are “the same.” For example, we discussed the fact that a conditional and its contrapositive have the same logical content. However, the equals sign (=) has already got a job...Some logical statements are “the same.” For example, we discussed the fact that a conditional and its contrapositive have the same logical content. However, the equals sign (=) has already got a job; it is used to indicate that two numerical quantities are the same. The formal definition of logical equivalence is two compound sentences are logically equivalent if in a truth table, the truth values of the two sentences are equal in every row. Thus, we use the symbol (≅) instead.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_(Levin)/3%3A_Symbolic_Logic_and_Proofs/3.1%3A_Propositional_LogicA proposition is simply a statement. Propositional logic studies the ways statements can interact with each other. It is important to remember that propositional logic does not really care about the c...A proposition is simply a statement. Propositional logic studies the ways statements can interact with each other. It is important to remember that propositional logic does not really care about the content of the statements. For example, in terms of propositional logic, the claims, “if the moon is made of cheese then basketballs are round,” and “if spiders have eight legs then Sam walks with a limp” are exactly the same. They are both implications.
- https://math.libretexts.org/Courses/Mount_Royal_University/Mathematical_Reasoning/1%3A_Basic_Language_of_Mathematics/1.1%3A_Compound_StatementsWe can make a new statement from old statements; we call these compound propositions or compound statements.
- https://math.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame_IN/SMC%3A_MATH_339_-_Discrete_Mathematics_(Rohatgi)/Text/3%3A_Symbolic_Logic_and_Proofs/3.2%3A_Propositional_LogicA proposition is simply a statement. Propositional logic studies the ways statements can interact with each other. It is important to remember that propositional logic does not really care about the c...A proposition is simply a statement. Propositional logic studies the ways statements can interact with each other. It is important to remember that propositional logic does not really care about the content of the statements. For example, in terms of propositional logic, the claims, “if the moon is made of cheese then basketballs are round,” and “if spiders have eight legs then Sam walks with a limp” are exactly the same. They are both implications.
- https://math.libretexts.org/Courses/Stanford_Online_High_School/Logic_for_All%3A_An_Introduction_to_Logical_Reasoning/09%3A_More_on_Truth_TablesThis page explores truth tables and their importance in logical reasoning, covering propositional connectives, construction of truth tables, and concepts such as tautologies, contradictions, and conti...This page explores truth tables and their importance in logical reasoning, covering propositional connectives, construction of truth tables, and concepts such as tautologies, contradictions, and contingencies. It illustrates their practical applications in validating arguments, circuit design, software testing, and legal analysis while addressing misconceptions.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Proofs_and_Concepts_-_The_Fundamentals_of_Abstract_Mathematics_(Morris_and_Morris)/01%3A_Propositional_Logic/1.06%3A_Tautologies_and_contradictionswhereas if \(P\) and \(Q\) are true, but \(R\) is false, then \[\begin{aligned} \bigl( P \& (\lnot Q \lor \lnot R) \bigr) \Rightarrow (P \Rightarrow \lnot Q) & \quad = \quad \bigl( \mathsf{T} \& (\lno...whereas if \(P\) and \(Q\) are true, but \(R\) is false, then \[\begin{aligned} \bigl( P \& (\lnot Q \lor \lnot R) \bigr) \Rightarrow (P \Rightarrow \lnot Q) & \quad = \quad \bigl( \mathsf{T} \& (\lnot \mathsf{T} \lor \lnot \mathsf{F}) \bigr) \Rightarrow (\mathsf{T} \Rightarrow \lnot \mathsf{T}) \\& \quad = \quad \bigl( \mathsf{T} \& (\mathsf{F} \lor \mathsf{T}) \bigr) \Rightarrow (\mathsf{T} \Rightarrow \mathsf{F}) \\& \quad = \quad \bigl( \mathsf{T} \& \mathsf{T} \bigr) \Rightarrow (\mathsf{F…
