(10) Draw a truth table for the compound statementq∨p→((p∧q)∨(q∧r))" role="presentation"...

Question

1. (10) Draw a truth table for the compound statementq∨p→((p∧q)∨(q∧r))" role="presentation" style="display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">q∨p→((p∧q)∨(q∧r))
2. (15) Use the laws of logic to show that((s∨r∨¬q)∧(¬s∨¬r∨s)∧(r∨¬s∨¬q))=(¬q∨r)" role="presentation" style="display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">((s∨r∨¬q)∧(¬s∨¬r∨s)∧(r∨¬s∨¬q))=(¬q∨r)
3. (30) Accepting the following premises, conclude that it rained. (Hint:Remember that you can use laws of logic as well as rules of inference.)
  1. It rained if it was cloudy and humid.
  2. Either the sun was out or it was cloudy.
  3. Either it was humid or it was hot.
  4. If it was hot, the sun was out.
  5. The sun was not out.
4. (5) The premises in problem 3 don't make sense in the real world. Why?
5. (30) Show by twice-applied universal generalization that((A−D)∪((B∪C)−D))=((A∪B∪C)−D)" role="presentation" style="display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">((A−D)∪((B∪C)−D))=((A∪B∪C)−D)
6. (10)Disproveby counter-example that(A∩C)∪(B∩D)⊆((A∩C)∪B)∩D" role="presentation" style="display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">(A∩C)∪(B∩D)⊆((A∩C)∪B)∩D

Vaibhav · Accepted Answer

Solution 1: q∨p→((p∧q)∨(q∧r))
					Truth Table
	p
	q
	r
	p∧q
	q∧r
	q∨p
	(p∧q)∨(q∧r)
	q∨p→((p∧q)∨(q∧r))
	T
	T
	T
	T
	T
	T
	T
	T
	T
	T
	F
	T
	F
	T
	T
	T
	T
	F
	T
	F
	F
	T
	F
	F
	T
	F
	F
	F
	F
	T
	F
	F
	F
	T
	T
	F
	T
	T
	T
	T
	F
	T
	F
	F
	F
	T
	F
	F
	F
	F
	T
	F
	F
	F
	F
	T
	F
	F
	F
	F
	F
	F
	F
	T
Solution 2: ((s∨r∨¬q)∧(¬s∨¬r∨s)∧(r∨¬s∨¬q))=(¬q∨r)
Considering LHS, 
(1) ((s∨r∨¬q)∧(¬s∨¬r∨s)∧(r∨¬s∨¬q)),
Using commutativity, ¬r∨s =s∨¬r,
(s∨r∨¬q)∧(¬s∨s∨¬r)∧(r∨¬s∨¬q))
Because ¬s∨s = T, Therefore, LHS reduces to,
(s∨r∨¬q)∧(T∨¬r)∧(r∨¬s∨¬q))
Now, using Domination Law, (T∨¬r) = T,
(s∨r∨¬q)∧(T)∧(r∨¬s∨¬q)),
Using Identity law, (s∨r∨¬q)∧(T) = (s∨r∨¬q). Hence,
(s∨r∨¬q)∧(r∨¬s∨¬q))
By Distributive law,

(10) Draw a truth table for the compound statementq∨p→((p∧q)∨(q∧r))" role="presentation" style="display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap;...

Answer To: (10) Draw a truth table for the compound statementq∨p→((p∧q)∨(q∧r))" role="presentation"...

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