Question

Simplify: ∼(p→q)∧((∼p∧q)∨∼(p∨q))=F.

a) ∼p→(p∨∼(p∨q))=p∨q
b) ∼p→(p∨∼(p∨q))=F
c) ∼p→(p∨∼p)=F
d) ∼p→(p∨∼(p∨q))=∼(p∧q)

Answer 1

To simplify the given expression, ∼(p→q)∧((∼p∧q)∨∼(p∨q)), we can break it down step by step. By applying logical equivalences and properties like the distributive property, we can simplify the expression to ∼p∨(q∧∼(p∨q)).

Step-by-step explanation:

To simplify the expression ∼(p→q)∧((∼p∧q)∨∼(p∨q)), we can break it down step by step:

1. Step 1: Simplify ∼(p→q)
2. Using the logical equivalence of ¬(p→q) = p∧¬q, we can rewrite this as ∼(p∧¬q).
3. Step 2: Simplify (∼p∧q)∨∼(p∨q)
4. Using the distributive property of ∧ over ∨, we can expand this expression to (∼p∨∼(p∨q))∧(q∨∼(p∨q)).
5. Step 3: Simplify (∼p∨∼(p∨q))∧(q∨∼(p∨q))
6. Using the distributive property again, we can rewrite the expression as (∼p∧q)∨(∼p∧∼(p∨q))∨(q∧q)∨(q∧∼(p∨q)).
7. Step 4: Simplify ∼(p∧q)
8. Using the logical equivalence of ¬(p∧q) = ¬p∨¬q, we can rewrite this as (∼p∨∼q).
9. Step 5: Simplify (∼p∧q)∨(∼p∧∼(p∨q))∨(q∧q)∨(q∧∼(p∨q))
10. By applying the identity law of ∧ and the domination law of ∨, we can simplify this to (∼p∨∼q)∨(q∧∼(p∨q)).
11. Step 6: Simplify (∼p∨∼q)∨(q∧∼(p∨q))
12. Using the logical equivalence of p∨(q∧∼p) = (p∨q)∧(p∨∼p), we can rewrite this as ∼p∨(q∧∼(p∨q)).

Therefore, the simplified expression is ∼p∨(q∧∼(p∨q)).