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Chapter 3: Problem 15

Let \(p\) be the statement "Ram races", \(q\) be the statement "Ram wins". The statement \(\sim(p \vee \sim q)\) is : (1) Ram does not race and Ram does not win. (2) It is not true that Ram races or Ram does not win. (3) Ram does not race and Ram wins (4) It is not true that Ram races and Ram does not win.

Short Answer

Expert verified
Option (3): Ram does not race and Ram wins

Step by step solution

01

Understand the Given Statement

The statement to simplify and interpret is \(eg (p \vee \eg q)\). Here, \(p\) represents 'Ram races', and \(q\) represents 'Ram wins'.
02

Apply De Morgan's Laws

De Morgan's Laws help simplify logical expressions involving negations. The statement \(eg (p \vee \eg q)\) can be rewritten using De Morgan's Laws: \(eg p \wedge \eg(eg q)\).
03

Simplify the Expression Further

The expression \(eg p \wedge \eg(eg q)\) simplifies to \(eg p \wedge q\). This means 'Ram does not race' and 'Ram wins'.
04

Match with Given Options

Compare the simplified expression \(eg p \wedge q\) with the given options. Option (3) states 'Ram does not race and Ram wins', which matches our simplified expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negation in logic
In logical reasoning, negation is a fundamental operation. Negation simply means reversing the truth value of a statement.
If a statement is true, its negation will be false, and vice versa. For a statement represented as 'p', its negation is written as '¬p' (or not p).
More formally:
- If 'p' is true, then '¬p' (not p) is false.
- If 'p' is false, then '¬p' (not p) is true.
In the given exercise, the initial statement to simplify involves a negation: ¬(p ∨ ¬q). Understanding how negation works is key to breaking down and interpreting this statement accurately.
De Morgan's Laws
De Morgan's Laws are essential principles in logical reasoning that help simplify expressions involving negations and conjunctions (AND) or disjunctions (OR).
These laws state:
- The negation of a disjunction: ¬(p ∨ q) = (¬p) ∧ (¬q)
- The negation of a conjunction: ¬(p ∧ q) = (¬p) ∨ (¬q)
In the exercise, the statement ¬(p ∨ ¬q) is given. Applying De Morgan's Law, we can rewrite it as (¬p) ∧ (¬(¬q)).
By simplifying ¬(¬q) to q, we get (¬p) ∧ q. This simplification makes the expression more manageable and easier to understand.
Finally, this means 'Ram does not race' and 'Ram wins', which we later match to one of the options.
Logical conjunctions
A logical conjunction is denoted by the symbol ∧, which translates to 'AND' in everyday language. It means both conditions must be true for the entire statement to be true.
For example, if p and q are two statements, 'p ∧ q' is true only if both p and q are true.
In the simplified expression (¬p) ∧ q from our problem, it means 'Ram does not race' (¬p) and 'Ram wins' (q).
- If p (Ram races) is false, ¬p (Ram does not race) is true.
- If q (Ram wins) is true, then combining these using 'AND' gives us a true statement.
Therefore, understanding conjunctions helps clarify why the simplified statement matches option (3): 'Ram does not race, and Ram wins'.

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