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Verify if the following proposition is valid:
Verify if the following proposition is valid:
$( P => Q ) \wedge ( P => R ) = P => ( Q \wedge R )$
Answer
Answer
AITruth table for $(P \Rightarrow Q) \wedge (P \Rightarrow R)$ and $P \Rightarrow (Q \wedge R)$ (verified with the boolean tool):
The two final columns $(P \Rightarrow Q) \wedge (P \Rightarrow R)$ and $P \Rightarrow (Q \wedge R)$ are identical in every row. Hence the given proposition is VALID (the two sides are logically equivalent, a tautology of the biconditional).
| P | Q | R | P=>Q | P=>R | (P=>Q).(P=>R) | Q.R | P=>(Q.R) |
| 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
From ISC 2023 Specimen Computer Science Paper 1, question 5(ii).