~(𝑷𝜦~𝑸) from inner to outer has parts 𝑷, 𝑸, ~𝑸, 𝑷𝜦~𝑸, ~(𝑷𝜦~𝑸)
Now make a chart for lists of T's and F's under each part:
| 𝑷 | 𝑸 | ~𝑸 | 𝑷𝜦~𝑸 | ~(𝑷𝜦~𝑸)|
| | | | | |
| | | | | |
| | | | | |
| | | | | |
Start by putting TTFF under the first letter P and TFTF under the second
letter Q
| 𝑷 | 𝑸 | ~𝑸 | 𝑷𝜦~𝑸 | ~(𝑷𝜦~𝑸)|
| T | T | | | |
| T | F | | | |
| F | T | | | |
| F | F | | | |
Now fill in the column under ~𝑸 according to the rule, if 𝑸 on that line
has T, put F under ~𝑸 on that line, If 𝑸 on that line has F, put T under
~𝑸.
| 𝑷 | 𝑸 | ~𝑸 | 𝑷𝜦~𝑸 | ~(𝑷𝜦~𝑸)|
| T | T | F | | |
| T | F | T | | |
| F | T | F | | |
| F | F | T | | |
𝑷𝜦~𝑸 is made up of 𝑷, ~𝑸, and 𝜦 between them. Rule: if 𝑷 and ~𝑸 both
have T's on that line, then put T under 𝑷𝜦~𝑸, but if either one is F, then
put F.
| 𝑷 | 𝑸 | ~𝑸 | 𝑷𝜦~𝑸 | ~(𝑷𝜦~𝑸)|
| T | T | F | F | |
| T | F | T | T | |
| F | T | F | F | |
| F | F | T | F | |
Now fill in the column under ~(𝑷𝜦~𝑸) according to the rule, if 𝑷𝜦~𝑸 on that
line has T, put F under ~(𝑷𝜦~𝑸) on that line, If 𝑷𝜦~𝑸 on that line has F,
put T under ~(𝑷𝜦~𝑸)
| 𝑷 | 𝑸 | ~𝑸 | 𝑷𝜦~𝑸 | ~(𝑷𝜦~𝑸)|
| T | T | F | F | T |
| T | F | T | T | F |
| F | T | F | F | T |
| F | F | T | F | T |
Edwin
