Discrete Mathematics | Propositional Logic | Test 1 | Question: 3

Question

Statements $P$ and $Q$ are said to be logically equivalent if they have the same truth value in every model. Now, Consider the following statements:   
      
i. Sentences $\textit{A provided B}$ and $\textit{(not A) or B}$ are $\textit{logically equivalent}$      
      
ii. Sentences $\textit{if A and B then C}$, $\textit{C provided (A and B)}$, and $\textit{(A and B) only if C}$ are logically equivalent      
       

• A. Only $(i)$ is correct     
        
• B. Only $(ii)$ is correct     
        
• C. Both $(i)$ and $(ii)$ are correct     
         
• D. None of the above

Answer 1

Option B is correct.

Let’s Discuss each Statement,

Statement 1: A provided B which means A if B and logically expressed as B → A 

                      B → A  ≡ ~B v A 

                    which can be translated as not B or A but in Statement it is saying not A or B which is logically expressed as ~ A v B ≡ A → B

                  And A → B which is not equivalent to B → A 

 Hence, this statement is Incorrect.

Statement 2.  If A and B then C which is logically expressed as  (A ∧ B) → C 

                       C provided A and B which means C if A and B which also means if A and B then C and can also be expressed as  (A ∧ B) → C 

                       A and B only if C which logically expressed as   (A ∧ B) → C.

as all three sub-statements are expressing the same thing which is  (A ∧ B) → C.

Hence, this statement is Correct.

[ Note: “~” symbol means “Negation” ]

[Tip:  remember this thing if there is “if” between two propositions then use the “Backward Implication symbol” for example A if B means B → A

and if there is “only if” between two propositions then use “forward Implication” for example A only if B means A → B.]