Take the statement: "Either Brown is in Barcelona or Jones owns a Ford." I know that if one element of the conjunction is true (Jones owns a Ford, for example), then the whole thing is true. Does the same thing apply for falsity? If one element is false, is the whole thing false?
If one element of a conjunction is false, is the whole statement false?
Yes. A conjunction of two propositions is only true when BOTH propositions constituting the conjunction are true.
This is a technical point but instead of "element" the term you are asking about is the "operand". In a conjunction statement such as "φ ∧ ψ" there are three elements, two operands (φ and ψ) and the conjunction (or connective) operator (∧). Your question could also be phrased, "If one [statement, proposition, assertion, premise, etc.] in a conjunction..." (Note that the ampersand is not as common anymore, but along with ∧ and •, & is an acceptable symbol for the "and" operator of a conjunction.)
If one element of a [disjunction] is false, is the whole statement false?
The answer is different if you mean an inclusive disjunction or an exclusive disjunction. In both cases they are false when both operands are false, but the exclusive disjunction is also false when both operands are true. (See the truth table below)
Take the statement: "Either Brown is in Barcelona or Jones owns a Ford."
First of all, that statement is not a conjunction statement, it is a disjunction. This is the truth table for disjunction, "inclusive" - a more common "or" statement" - is on the left and "exclusive" disjunction is on the right:
To be clear, the "aVb" column is "inclusive or" and the "a V̲ b" column is "exclusive or" (sometimes indicated by "xor").
I know that if one element of the conjunction(sic) is true (Jones owns a Ford, for example), then the whole thing is true.
Given your example is a disjunction, I am presuming you meant your question about disjunctions.
Does the same thing apply for falsity? If one element is false, is the whole thing false?
Yes for conjunctions.
No for disjunctions.
Although in terms of grammar your sentence uses a conjunction to join the two clauses, in terms of a type of logical complex sentence, it is a DISJUNCTION. This is because it uses the connective or, not the connective and.
As you rightly say, if one disjunct in the disjunction is true then the whole sentence is true. On that basis, therefore, the sentence must be true if one of the disjuncts is true and the other one is false. So, no, it is not the case that if one disjunct in a disjunction is false the sentence is necessarily false.