Question

The statement “If $ {2^2} = 5 $ then I get first class” is logically equivalent to
 A. $ {2^2} = 5 $ and I do not get first class
B. $ {2^2} = 5 $ or I do not get first class
C. $ {2^2} \ne 5 $ or I get first class
D. None of these

Answer 1

Hint: The statement has if and then. So first divide the statement into two other smaller statements and label them as p, q respectively. When we represent the given statement logically we get the form $ p \to q $ , if p then q. This statement is called a conditional statement. If a statement is in the form $ p \to q $ , then it can also be written as $ \neg p \vee q $ . The logically equivalent statement of $ p \to q $ will have a not p statement and q statement combined with “or”.

Complete step-by-step answer:
 True and false are called truth values. A statement of the form “if p then q” or “p implies q” is called a conditional statement.
We are given to find the logically equivalent statement of the statement “If $ {2^2} = 5 $ then I get first class”.
Let us first divide the statement into two.
Let $ {2^2} = 5 $ be p and “I get first class” be q.
Then the statement “If $ {2^2} = 5 $ then I get first class” logically looks like $ p \to q $ , if p then q.
When a statement is of the form $ p \to q $ , then it is also logically equivalent to $ \neg p \vee q $
 $ \neg p $ is not p, this means not $ {2^2} = 5 $ which is $ {2^2} \ne 5 $
‘˅’ is a symbol used to represent “or”.
Therefore, $ \neg p \vee q $ is “ $ {2^2} \ne 5 $ or I get first class”
“If $ {2^2} = 5 $ then I get first class” is logically equivalent to “ $ {2^2} \ne 5 $ or I get first class”
So, the correct answer is “Option C”.

Note: Negation of a statement containing ‘or’ becomes a statement containing ‘and’ and negation of a statement containing ‘and’ becomes a statement containing ‘or’. Negation (logical complement) of a negation results in the original state of the object or statement. Negation is the opposite; negation of negation is the opposite of opposite. Here $ p \to q $ is logically equivalent to $ \neg p \vee q $ because their truth values are the same.