Question

Find the compound statement “It is snowing and it is not that I am cold”. If \[p\] : it is snowing, \[q\] : I am cold.
A. \[p \wedge \left( { \sim q} \right)\]
B. \[p \wedge q\]
C. \[\left( { \sim p} \right) \wedge q\]
D. \[\left( { \sim p} \right) \wedge \left( { \sim q} \right)\]
E. \[p \vee \left( { \sim q} \right)\]

Answer 1

Hint: Find the negation of the given statement \[p\] : it is snowing, and \[q\] : I am cold. After that join the two statements with an AND to get the required answer.

Formula used:
\[P \wedge Q\] means P and Q.
\[P \vee Q\] means P or Q.
\[\sim p\] means p is false.

Complete step by step solution
From the given information, we get:
\[p\]: It is snowing
\[q\]: I am cold
Now,
We are asked:
It is snowing and it is not that I am cold
For this,
We have:
For “It is not that I am cold”: \[ \sim q\]
The two statements are connected with an AND. So,
The answer becomes:
\[p \wedge \left( { \sim q} \right)\]

The correct answer is option A.

Additional information:
Mathematical statements that can divided into two or more basic statements are called compound statements.
Every statement is either true or false. The logical symbols are \[\wedge\], \[\vee\], \[\sim\], \[ \rightarrow\], \[ \leftrightarrow \].
Negation: If \[P\] is true, then \[\sim P\] is false.
Conjunction: \[P\wedge Q\] should be true, if both P and Q are true otherwise false.
Disjuction: \[P \vee Q\] is true if either P or Q true. \[P \vee Q\] is false, if P and Q both are false.
Implication: \[P \leftrightarrow Q\] means that P and Q are equivalent. \[P \leftrightarrow Q\] is true if P and Q both are true or P and Q both are false.

Note: Students often confused with \[\wedge\] and \[\vee\]. The meaning of \[\wedge\] is “AND”. The meaning of \[\vee\] is “OR”.