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Question: The statement “If \( {2^2} = 5 \) then I get first class” is logically equivalent to A. \( {2^2} ...

The statement “If 22=5{2^2} = 5 then I get first class” is logically equivalent to
A. 22=5{2^2} = 5 and I do not get first class
B. 22=5{2^2} = 5 or I do not get first class
C. 22≠5{2^2} \ne 5 or I get first class
D. None of these

Explanation

Solution

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→qp \to q , if p then q. This statement is called a conditional statement. If a statement is in the form p→qp \to q , then it can also be written as ¬p∨q\neg p \vee q . The logically equivalent statement of p→qp \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 22=5{2^2} = 5 then I get first class”.
Let us first divide the statement into two.
Let 22=5{2^2} = 5 be p and “I get first class” be q.
Then the statement “If 22=5{2^2} = 5 then I get first class” logically looks like p→qp \to q , if p then q.
When a statement is of the form p→qp \to q , then it is also logically equivalent to ¬p∨q\neg p \vee q
¬p\neg p is not p, this means not 22=5{2^2} = 5 which is 22≠5{2^2} \ne 5
‘˅’ is a symbol used to represent “or”.
Therefore, ¬p∨q\neg p \vee q is “ 22≠5{2^2} \ne 5 or I get first class”
“If 22=5{2^2} = 5 then I get first class” is logically equivalent to “ 22≠5{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→qp \to q is logically equivalent to ¬p∨q\neg p \vee q because their truth values are the same.