How to solve a Logic problem?

Question

Use a truth table or Venn diagram to show that the statement p v (q ^ r) is equivalent to (p v q) ^ (p v r)

Show that this statement is not equivalent to (p v q) ^ r.

 

Be sure to explain your answer and don't just provide a truth table or Venn diagram.  Why does your table or diagram verify the result?

Use a truth table or Venn diagram to show that the statement p v (q ^ r) is equivalent to (p v q) ^ (p v r)

Show that this statement is not equivalent to (p v q) ^ r.

 

 

Answer 1

Let q={ 0 1 4 9 16 25 36 }, r={ 1 3 5 7 9 11 }, p={ 0 2 4 6 8 10 }

So q ^ r={ 1 9 }, p v q={ 0 1 2 4 6 8 9 10 16 25 36 }, p v r={ 0 1 2 3 4 5 6 7 8 9 10 11 }

(p v q) ^ (p v r)={ 0 1 2 4 6 8 9 10 }

p v (q ^ r)={ 0 1 2 4 6 8 9 10 }, so (p v q) ^ (p v r) = p v (q ^ r)

(p v q) ^ r={ 1 9 } ≠ (p v q) ^ (p v r).

Draw two intersecting circles representing sets q and r. Where they intersect is q ^ r (in the example the intersection would contain the numbers 1 and 9. Now consider augmenting the sets by the contents of p (this is the union of p with each of the two intersecting sets). This time the intersection would contain all the elements of p as well as 1 and 9. This demonstrates the first part of the question.

But if the p elements are added only to q then the intersection only contains 1 9 because there are no more common elements in r than there were before. This demonstrates the second part of the question.