- 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 $(p)$