Prove Distributive laws
1. p and (q or r) = (p and q) or (p and r)
Solution:
To prove `p^^(qvvr)=(p^^q)vv(p^^r)`, we have to first prepare the following truth table
| `(1)` | `(2)` | `(3)` | `(4)=(2)vv(3)` | `(5)=(1)^^(4)` | `(6)=(1)^^(2)` | `(7)=(1)^^(3)` | `(8)=(6)vv(7)` |
| `p` | `q` | `r` | `qvvr` | `p^^(qvvr)` | `p^^q` | `p^^r` | `(p^^q)vv(p^^r)` |
| T | T | T | T | T | T | T | T |
| T | T | F | T | T | T | F | T |
| T | F | T | T | T | F | T | T |
| T | F | F | F | F | F | F | F |
| F | T | T | T | F | F | F | F |
| F | T | F | T | F | F | F | F |
| F | F | T | T | F | F | F | F |
| F | F | F | F | F | F | F | F |
from this table, we can say that columns (5) and (8) are identical.
`:. p^^(qvvr)=(p^^q)vv(p^^r)`
2. p or (q and r) = (p or q) and (p or r)
Solution:
To prove `pvv(q^^r)=(pvvq)^^(pvvr)`, we have to first prepare the following truth table
| `(1)` | `(2)` | `(3)` | `(4)=(2)^^(3)` | `(5)=(1)vv(4)` | `(6)=(1)vv(2)` | `(7)=(1)vv(3)` | `(8)=(6)^^(7)` |
| `p` | `q` | `r` | `q^^r` | `pvv(q^^r)` | `pvvq` | `pvvr` | `(pvvq)^^(pvvr)` |
| T | T | T | T | T | T | T | T |
| T | T | F | F | T | T | T | T |
| T | F | T | F | T | T | T | T |
| T | F | F | F | T | T | T | T |
| F | T | T | T | T | T | T | T |
| F | T | F | F | F | T | F | F |
| F | F | T | F | F | F | T | F |
| F | F | F | F | F | F | F | F |
from this table, we can say that columns (5) and (8) are identical.
`:. pvv(q^^r)=(pvvq)^^(pvvr)`
This material is intended as a summary. Use your textbook for detail explanation.
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