Question

Give a formal proof for the following tautology by using the CP rule.

A v B→(¬ B →(A ^ ¬ B))

Answer 1

Solution for the problem is provided below, please comment if any doubts:

Note: Since the solution contains equations, to avoid format loss, U added the screenshot of the solution, raw data is also included at the end

Raw data:

The procedure to prove the tautology using conditional proof (CP) rules is following the procedure, From P, derive Q, then P→Q, P is the LHS and Q is the RHS.

Here P=> A v B

And Q => (¬ B →(A ^ ¬ B))

So from “A v B” we need to derive “(¬ B →(A ^ ¬ B))” to prove the tautology

1. A v B                       :Given premise(P)
2. ¬ B                           : Given premise derived(P)[To get (¬ B →(A ^ ¬ B) True]
3. A                              :1, 2, Disjunctive Syllogism(DS) rule
4. A ^ ¬ B                    :2, 3, Conjunction
5. (¬ B →(A ^ ¬ B))    :2, 4, CP rule

QED                            :1-5 CP rule

Hence prove the tautology, “A v B→(¬ B →(A ^ ¬ B))” using CP rule