Question

This assignment deals with Logical Equivalences. Review section 1.7 of the text before completing the assignment. The assignment may be handed in twice before it is graded. Consider the statements in the left column of the tables below. Translate each into a propositional statement. In the box below, indicate which two statements are logically equivalent. The gray shaded box is the Equation editor that should be used to enter the propositional expression.
Question 1
Statement Reason
Whenever there is a puppy in the house, I feel happy
If I am happy, then there is a puppy in the house
If there is not a puppy in the house, then I am not happy.
If I am not happy, then there is no puppy in the house
Question 2
Statement Reason
If I am in school today, then I am in CSC231 class
If I am not in school today, then I am not civics class
If I am not in CSC231 class, then I am not in school today
If I am in CSC231 class, then I am in school today

Answer 1

Answer:

Question (1) the statements (i) and( iv) are logically equivalent and statements (ii) and (iii) are logically equivalent. Question (2) the statements (i) and (iii) are logically equivalent.

Explanation:

Solution

Question (1)

Now,

Lets us p as puppy in the house, and q as i am happy

So,

p : puppy in the house,  and q : i am happy

Thus,

The Statements

(i) so if there is a puppy in the house, I feel happy :  p -> q

(ii) If I am happy, then there is a puppy in the house : q -> p

(iii) If there is no puppy in the house, then I am not happy.   : ~ p -> ~q

(iv) If I am not happy, then there is no puppy in the house : ~q -> ~p

Hence, the statements (i) and( iv) are logically equivalent and statements (ii) and (iii) are logically equivalent.

Question (2)

Let us denote p as i am in school today, and q as i am in CSC231 class, and r as i am in civics class,

So,

p: i am in school today, q: i am in CSC231 class, r: i am in civics class,

Now,

(i) if I am in school today, then I am in CSC231 class  :p -> q

(ii) If I am not in school today, then I am not civics class  :~p -> ~r

(iii) If I am not in CSC231 class, then I am not in school today  :~q -> ~p

(iv) If I am in CSC231 class, then I am in school today  : q -> p

Therefore, the statements i) and iii) are logically equivalent.