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Implication

Definition 2.3.13   Let $ p$ and $ q$ be two propositions. Then the truth table of $ p
\Longrightarrow q$ is as follows:
$ p$ $ q$ $ p
\Longrightarrow q$
T T T
T F F
F T T
F F T

This means that by a correct argument, when the premice is true, then the conclusion is true. When the premice is false, you can get any conclusion, even with a correct argument. For example the sentence ``Jerusalem is Pakistan's capital,thus an apple has wings'' is logically true.

Proposition 2.3.14   Let $ p$,$ q$ and $ r$ are three propositions, then $ (p \Rightarrow q )
\wedge (q \Rightarrow r )$ implies $ p \Rightarrow r $.

Proof. As usual, we use a truth table.
$ p$ $ q$ $ r$ $ p \Rightarrow q$ $ q \Rightarrow r$ $ (p \Rightarrow q )
\wedge (q \Rightarrow r )$ $ p \Rightarrow r $ $ [(p
\Rightarrow q) \wedge (q \Rightarrow r)]\Longrightarrow (p \Rightarrow
r)$
T T T T T T T T
T T F T F F F T
T F T F T F T T
T F F F T F F T
F T T T T T T T
F T F T F F T T
F F T T T T T T
F F F T T T T T
The last column shows that we have here a tautology. $ \qedsymbol$

Proposition 2.3.15   Let $ p$ and $ q$ be two propositions. Then $ p
\Longrightarrow q$ is logically equivalent to the statement $ \overline{p} \vee q$.

The proof has to be done with a truth table, which we leave to the reader. This proposition will reveal useful for finding the negation of an implication. For example:

Example 2.3.16   In Calculus, we learn the definition of a continous function at a point $ x_0$. A function $ f$, defined on a neighborhood of a real $ x_0$ is continuous at $ x_0$ if the following holds:

$\displaystyle \forall \varepsilon >0, \; \exists \alpha > 0 \; \vert \; \vert x-x_0\vert<\alpha
 \Longrightarrow \vert f(x)-f(x_0)\vert<\varepsilon.$    

The negation of this, i.e. the fact that $ f$ is discontinuous at $ x_0$ is described by the following statement:

$\displaystyle \exists \varepsilon >0 \; \vert \; \forall \alpha >0, \; \exists ...
...<\alpha \\  \text{and}\\  \vert f(x)-f(x_0)\vert \geq \varepsilon.
 \end{cases}$    


next up previous contents
Next: Equivalence Up: Logical connectors Previous: Tautologies and contradictions   Contents
root 2002-06-10