Question

Prove p ∨ (q ∧ r) ⇒ (p ∨ q) ∧ (p ∨ r) by constructing a proof tree whose premise is p∨(q∧r) and whose conclusion is (p∨q)∧(p∨r).

Answer 1

Prove p ∨ (q ∧ r) ⇒ (p ∨ q) ∧ (p ∨ r) by constructing a proof tree whose premise is p∨(q∧r) and whose conclusion is (p∨q)∧(p∨r) as following. Image form is also enclosed in attachment.

1.((p∨(q∧r))→((p∨q)∧(p∨r)))

2.(p∨(q∧r))

3.¬((p∨q)∧(p∨r))

4.¬(p∨q)(3)

6.¬p(4)

7.¬q(4)

8.p (2)
x

9.(q∧r)(2)

10.q(9)

11.r (9)
x

5.¬(p∨r)(3)

12.¬p(5)

13.¬r(5)

14.p (2)
x

15.(q∧r)(2)

16.q(15)

17.r
x