Prove De-Morgan's laws
1. not(p or q) = not p and not qSolution:To prove `~(pvvq)=~p^^~q`, we have to first prepare the following truth table
| `(1)` | `(2)` | `(3)=(1)vv(2)` | `(4)=~(3)` | `(5)=~(1)` | `(6)=~(2)` | `(7)=(5)^^(6)` |
| `p` | `q` | `pvvq` | `~(pvvq)` | `~p` | `~q` | `~p^^~q` |
| T | T | T | F | F | F | F |
| T | F | T | F | F | T | F |
| F | T | T | F | T | F | F |
| F | F | F | T | T | T | T |
from this table, we can say that columns (4) and (7) are identical.
`:. ~(pvvq)=~p^^~q`
2. not(p and q) = not p or not qSolution:To prove `~(p^^q)=~pvv~q`, we have to first prepare the following truth table
| `(1)` | `(2)` | `(3)=(1)^^(2)` | `(4)=~(3)` | `(5)=~(1)` | `(6)=~(2)` | `(7)=(5)vv(6)` |
| `p` | `q` | `p^^q` | `~(p^^q)` | `~p` | `~q` | `~pvv~q` |
| T | T | T | F | F | F | F |
| T | F | F | T | F | T | T |
| F | T | F | T | T | F | T |
| F | F | F | T | T | T | T |
from this table, we can say that columns (4) and (7) are identical.
`:. ~(p^^q)=~pvv~q`
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