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$$\text{Equivalence}$$

$p \land T \equiv p\tag{Identity law 1}$

$$p\lor F \equiv p\tag {Identity law 2}$$

$$p\lor T \equiv T\tag{Domination law 1}$$

$$p\land F \equiv F\tag{Domination law 2}$$

So, in the above image, where the T and F are, I assume these represent something like:
T = "any true proposition"
F = "any false proposition"

And so the first row in English would be "If P and any true proposition is true then this is logically equivalent to P" is that right? I could imagine it being something a bit different.

Thanks in advance!

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2 Answers

You're basically right, though I would very slightly change the wording:

Let $T$ be a true proposition. Let $F$ be a false proposition.

$p \land T \equiv p$: We interpret this as:

The proposition "$p$ is true and $T$ is true" is equivalent to the proposition "$p$ is true."

$p \lor F \equiv p$: We interpret this as:

The proposition "$p$ is true or $F$ is false" is equivalent to the proposition "$p$ is true."

$p \lor T \equiv p$: We interpret this as:

The proposition "$p$ is true or $T$ is true" is equivalent to the proposition "$T$ is true."

$p \land F \equiv p$: We interpret this as:

The proposition "$p$ is true and $F$ is false" is equivalent to the proposition "$F$ is false."

To see why this works, write out the truth tables of each proposition and see that the equivalences do hold.

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Thanks! That was useful. –  papercuts Dec 14 '12 at 21:04
    
Would it be okay to replace "is equivalent to" with "if and only if"? –  papercuts Dec 14 '12 at 21:09
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@papercuts Yes. Those mean the same thing. (If it was useful please upvote! It's how we recognize good answers.) –  kigen Dec 14 '12 at 22:25
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  • Yes, $T$ = any proposition that is necessarily (or always) true (a tautology).
  • And $F$ = any proposition that is necessarily (or always) false (a contradiction).
  • The symbol $\;$"$\;\land\;$" denotes logical AND (conjunction).
  • The symbol $\;$"$\,\lor\,$" denotes logical OR (disjunction).
  • The symbol "$\;\equiv\,$" denotes "is logically equivalent to" or if you prefer, it denotes "if and only if".

*It might be helpful to review the truth-tables for the logical connectives $\land,\;\lor,\;\text{and}\;\equiv\;(\text{or}\;\iff)\;$ to understand why te following assertions must be true:

$$p \land T \equiv p$$ $$p \lor F \equiv p$$ $$p \lor T \equiv T$$ $$p \land F \equiv F$$

With respect to your second question.

Yes, for the first identity, we have that $\; p \land T\;$ is logically equivalent to $\; p.\;$
Put differently: $(p\,$ AND $\,T)\;$ if and only if $\;p$.

Since $T$ represents a tautology (true no matter what), then the truth-value of $\;p \land T\;$ depends only on the truth-value of $\;p\;$: When $p$ is false both sides of the equivalence are false, and when $p$ is true, both sides of the equivalence are true.

So yes,

$p \land T \equiv p$.

This also means $(p \land T \iff p):\quad$ ($p$ and $T$) if and only if $(p)$

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how did this not get any votes? +1 –  Amzoti May 3 '13 at 0:48
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