I was looking at an exercise where it asked the following:
$$\begin{array}{ccc} p&q&p\rightarrow q \\ T&T&T \\ &\ldots \end{array}$$
So, for the third column, I just put $T$ which was correct but I didn't understand what $\rightarrow$ meant. I have seen $\implies$ but I haven't the arrow. Are they the same thing?
Thanks a bunch!
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$\begingroup$The $\rightarrow$ symbol is a connective. It's a symbol which connects two propositions in the context of propositional logic (and its extensions, first-order logic, and so on).
The truth table of $\rightarrow$ is defined to be that $p\rightarrow q$ is false if and only if $p$ is true and $q$ is false.
Indeed this is the same meaning of $\implies$, but the difference is that $p\implies q$ is a statement about propositions, whereas $p\rightarrow q$ is a proposition. In some contexts, though, people don't make this distinction between material implication (the connective) and logical implication (the $\implies$ arrow). But they are not the same thing in every context of propositional logic.
$\endgroup$ 3 $\begingroup$It is a material conditional, or otherwise known as $p$ implies $q$, or if $p$, then $q$
The truth table for that is as follows
p q p implies q
T T T
T F F
F T T
F F T$\rightarrow$ can also be written as $\implies$.
In computer science, $p \implies q$ can be rewritten as (not p) or q, or !p||q
Given $p$, then we have $q$.
or $p$ implies $q$.
The two arrows mean the same thing.
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