Logic Notation question (specifically about logical equivalences) 
$$\text{Equivalence}$$
$p \land T \equiv p\tag{Identity law 1}$
$$p\lor F \equiv p\tag {Identity law 2}$$
$$p\lor T \equiv T\tag{Domination law 1}$$
$$p\land F \equiv F\tag{Domination law 2}$$
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!
 A: 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.
A: *

*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 $(p)$
