Some logical concerns

Theorems (major results), propositions (not so major results) and lemmas (minor results needed to derive propositions or theorems) are deduced from previously established theorems, propositions, lemmas and axioms (these are assumed without proof). We will use several logical concepts. We are usually trying to prove that $(A)$ implies $(B)$ or to show that $(A)$does not imply $(B)$. Here $(A)$ and $(B)$ are simple or complicated mathematical statements or formulae. To show that $(A)$ implies $(B)$, we have to prove that whenever $(A)$ is true, $(B)$ is also true. To show that $(A)$ does not imply $(B)$, we have to find one case where $(A)$ is true and $(B)$ is not. Thus to prove that ``every girl in New York City under the age of 5 has green eyes'' we could collect all the girls in New York City under the age of 5 in one room and look at the color of each girl's eyes. If we found one girl with blue eyes, then we would have shown that the statement ``every girl in New York City under the age of 5 has green eyes'' is false.

To facilitate communication we use a number of symbols ( $\Longrightarrow$, $\Longleftarrow$, $\Longleftrightarrow$) and give very specific meaning to some English phrases. The symbols `` $(A) \Longrightarrow (B)$'' and `` $(B) \Longleftarrow (A)$'' mean ``$(A)$ implies $(B)$'' and ``$(B)$ is implied by $(A)$,'' respectively, and are logically equivalent. Four other sentences that mean the same thing are:
1. $(B)$ if $(A)$.
2. $(A)$ only if $(B)$.
3. $(A)$ is sufficient for $(B)$.
4. $(B)$ is necessary for $(A)$.

Implication is transitive; that is, if $(A) \Longrightarrow (B)$ and $(B) \Longrightarrow (C)$, then $(A) \Longrightarrow (C)$. Every implication (for example, $(A) \Longrightarrow (B)$) has a converse obtained by interchanging the hypothesis and conclusion (in our example, $(B) \Longrightarrow (A)$) and a counterpositive ( $\mbox{not } (B) \Longrightarrow \mbox{ not }(A)$) obtained by replacing the hypothesis with the negation of the conclusion and the conclusion with the negation of the hypothesis. An implication and its converse need NOT have the same truth value, but an implication and its counterpositive always do. Try it with the statement ``every girl in New York City under the age of 5 has green eyes.''

The symbol $\Longleftrightarrow$ is a combination of $\Longrightarrow$ and $\Longleftarrow$ and denotes implication in both directions. Thus `` $(A) \Longleftrightarrow (B)$'' stands for ``$(A)$ if and only if $(B)$'' which we shall also write as ``$(A)$ iff $(B)$.''