Negation of the statement: $$\sqrt{5}$$ is an integer or 5 is irrational is:

We begin by translating the given English sentence into symbols. Let $$P$$ denote the statement “$$\sqrt{5}$$ is an integer.” Let $$Q$$ denote the statement “$$5$$ is irrational.” The original sentence is “$$\sqrt{5}$$ is an integer or 5 is irrational,” which is symbolically $$P \lor Q$$.

To find the negation, we recall the logical rule known as De Morgan’s law. The law states: for any two statements $$P$$ and $$Q$$, $$\neg(P \lor Q) \;=\; (\neg P) \land (\neg Q).$$ In words, “not (P or Q)” is equivalent to “(not P) and (not Q).”

Applying this formula to our case, we have $$\neg(P \lor Q) \;=\; (\neg P) \land (\neg Q).$$

Now we translate each piece back into ordinary language. The negation $$\neg P$$ is “$$\sqrt{5}$$ is not an integer.” The negation $$\neg Q$$ is “5 is not irrational,” i.e. “5 is rational” (indeed 5 is an integer, hence rational).

Combining these two with the connective “and,” we obtain “$$\sqrt{5}$$ is not an integer and 5 is not irrational.”

Examining the choices, this wording matches exactly with Option B.

Hence, the correct answer is Option B.