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Question 5 3 pts

Given proposition A: ( ¬( p ? q ) ? q ), and proposition B: ( ( ¬p ? ¬q ) ? q ), determine which type of law should be applied to show that the two propositions are logically equivalent.

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Question 8 3 pts

Let the domain for the variable x is the set of all integers. Is the quantified statement ? x (x*x=x) true or false?

Solution

Please follow the data and description :

1)

If p is false, q is false, and r is false, then the given expression p ^ (~q ^ r) is also false.

So the answer is OPTION B (False).

2)

If p is true, q is false, and r is true, then the given expression p -> (q -> r) is also true.

So the answer is OPTION A (True).

3)

De-Morgan\'s Law :

The negation of a conjunction or clearly a logical AND of the 2 statements is logically equivalent to the the disjunction logical OR of each of the statement\'s negation. Statements as \"not (A and B)\" is logically equivalent to \"not A or not B\". Similarly, the negation of a disjunction of 2 statements is logically equivalent to the conjunction of each statement\'s negation.

So the answer is OPTION B (De-Morgan\'s Law).

4)

Yes the statement p ^ q and the statement ~p ^ ~q are logically equivalent.

So the answer is OPTION A (True).

Hope this is helpful.