Since the conjunct of a true contingent proposition and a necessary proposition is contingent and hence it is contained in the set of all contingent propositions. Does that mean that the set of all contingent propositions the same set as the set of all true propositions?
No. Call T the set of true propositions and T* the set of true contingent propositions. Let P be a necessary proposition.
All necessary propositions are true, so P in T.
But no necessary proposition is contingent, because contingent just means neither necessary nor impossible. So P not in T*. Then T /= T*.
I have a guess about where your confusion comes from.
Let Q be a true contingent proposition. Obviously Q in T, Q in T*. Now consider the proposition "P and Q". Now "P and Q" is a true contingent proposition. So "P and Q" in T, "P and Q" in T*.
But "P and Q" is not the same as P, nor as Q. (The use of "the sky is blue and a = a" is nothing like the use of either "the sky is blue" nor "a = a".) So saying "P and Q" in T* does not imply P in T*.
So, in short, the answer is no, but I can understand the intuition behind this. The issue comes from the nature of each of these things; truth and possibly.
What is possible to tends to be divided into three branches:
1) that which is logically possible
2) that which is metaphysically possible
3) that which is physically possible
Whilst, on the other hand, truth is generally taken as a relation between language (or propositions) and reality, depending on which view you're taking. The more popular views on truth tend to be correspondence or deflationary views which equate the truth of a truth-bearer with that truth-bearing expressing something and that thing is the case. In each possible world, the set of true propositions will be specific to that world; each proposition will be true with respect to the reality that is the case in that possible world.
The sets of each of these things will be distinct, as the set of all possible propositions is far greater than the set of all true propositions. This is because the set of all contingent/possible propositions contains all the true propositions, as they must be possible if they are true. But the set also contains non-true propositions; i.e. "the Eiffel tower is in London" is possible on all three versions of possiblity mentioned above, yet it fails to fall into the set of true propositions.
It would be correct, concerning conjunction with necessary propositions, to claim that the set of all contingent propositions contains the set of all necessary propositions. But it wouldn't be the case that the set of all true propositions is identical to the set of all contingent propositions - the latter set is far greater.