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Alternatively, just the truth table method. Since each proposition must be equal to its matching variable, we can mark rows as contradictions if the two sets of columns are unequal. All rows without contradictions are valid answers to the question. I don't have to convince you of any given value of any given variable, because this is exhaustively examining every possible combination.
1	-	Alternatively, just the truth table method. Since each proposition must be equal to its matching variable, we can mark rows as contradictions if the two sets of columns are unequal. All rows without contradictions are valid answers to the question. I don't have to convince you of any given value of any given variable, because this is exhaustively examining every possible combination.
1	+	Let A, B, C, D, and E be five boolean variables.
2
3		Let P1, P2, P3, P4, and P5 be the five propositions listed in the question.
4
5		To find all possible answers, we can construct a truth table. Since each proposition must be equal to its matching variable, we can mark rows as contradictions if the two sets of columns are unequal. All rows without contradictions are valid answers to the question. I don't have to convince you of any given value of any given variable, because this is exhaustively examining every possible combination.
6
7		A B C D E \| P1 P2 P3 P4 P5 \| Contradiction?
8		----------+----------------+---------------
9		F F F F F \| T T F T F \| Yes
10		F F F F T \| F T F T F \| Yes
11		F F F T F \| F F F T F \| No
12		F F F T T \| F F F T F \| Yes
13		F F T F F \| F F F F F \| Yes
14		F F T F T \| F F F F F \| Yes
15		F F T T F \| F F F F F \| Yes
16		F F T T T \| F F F F F \| Yes
17		F T F F F \| F T F F F \| No
18		F T F F T \| F T F F F \| Yes
19		F T F T F \| F F F F F \| Yes
20		F T F T T \| F F F F F \| Yes
21		F T T F F \| F F T F F \| Yes
22		F T T F T \| F F T F F \| Yes
23		F T T T F \| F F T F F \| Yes
24		F T T T T \| F F T F F \| Yes
25		T F F F F \| T T F F F \| Yes
26		T F F F T \| F T F F F \| Yes
27		T F F T F \| F F F F F \| Yes
28		T F F T T \| F F F F F \| Yes
29		T F T F F \| F F F F F \| Yes
30		T F T F T \| F F F F F \| Yes
31		T F T T F \| F F F F F \| Yes
32		T F T T T \| F F F F F \| Yes
33		T T F F F \| F T F F F \| Yes
34		T T F F T \| F T F F F \| Yes
35		T T F T F \| F F F F F \| Yes
36	-	T T T T T \| F F T F T \| Yes
36	+
37		T T T F F \| F F T F F \| Yes
38		T T T F T \| F F T F F \| Yes
39		T T T T F \| F F T F F \| Yes
40		T T T T T \| F F T F T \| Yes
41
42		In conclusion, there are two possible answers:
43
44		(1) Only B is true.
45		(2) Only D is true.
46
47		They cannot be true at the same time.