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