FoldUnfold Table of Contents Tautologies and Contradictions Tautologies and Contradictions Definition: A formula is said to be a Tautology if every truth assignment to its component statements results in the formula being true. A formula is said to be a Contradiction if every truth assignment to its component statements results in the formula being false. It is easy … [Read more...]

## The Semantics of Propositional Logic

FoldUnfold Table of Contents The Semantics of Propositional Logic Negation Conjunction Disjunction Exclusive Disjunction / Exclusive Or Implication Biconditional The Semantics of Propositional Logic So far we have discussed propositional logical connectives and formulas. We noted that formulas derive from statements. We would now like to determine whether a formula is … [Read more...]

## Formulas in Propositional Logic

FoldUnfold Table of Contents Formulas in Propositional Logic Formulas in Propositional Logic We are now ready to properly define a formula in propositional logic. Definition: A Formula in propositional logic must satisfy one of the criteria below: a) A statement is a formula. b) If $\varphi$ is a formula then $\neg \varphi$ is a formula. c) If $\varphi$ and $\psi$ are … [Read more...]

## Propositional Logic Connectives

FoldUnfold Table of Contents Propositional Logic Connectives Propositional Logic Connectives Before we learn about propositional logic we must first understand the meaning of certain symbols which we will call connectives. We will use these connectives with statements. These statements will be denoted with the letters $P$, $Q$, and $R$, and they will state the … [Read more...]

## The AM-GM Inequality

FoldUnfold Table of Contents The AM-GM Inequality The AM-GM Inequality Theorem 1 (The AM-GM Inequality): Let $x_1, x_2, ..., x_n$ be nonnegative real numbers. Then $\displaystyle{\frac{x_1 + x_2 + ... + x_n}{n} \geq \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n}}$. "AM-GM" stands for "Arithmetic Mean - Geometric Mean". Proof: Consider the function $f(x) = -\ln x$. This … [Read more...]