Question

\(\sim(p \mathrm{~V} q) \mathrm{V}(\sim \mathrm{p} \Lambda \mathrm{q})=\ldots \ldots \ldots \ldots\) (a) q (b) \(\mathrm{p}\) (c) \(\sim p\) (d) \(\sim \mathrm{q}\)

Short answer

(c) ¬p

Step-by-step solution

1. Rewrite the given expression using logic symbols in place of their corresponding meanings.

The given expression is: \[\sim(p \lor q) \lor (\sim p \land q)\]

2. Apply De Morgan's law to the first term of the expression.

De Morgan's law states that: \[\sim(p \lor q) \equiv (\sim p) \land (\sim q)\] Therefore, the expression becomes: \[(\sim p \land \sim q) \lor (\sim p \land q)\]

3. Use the distributive law to simplify the expression.

The distributive law states that: \[p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)\] By the distributive law, our expression becomes: \[\sim p \land (\sim q \lor q)\]

4. Apply the law of excluded middle to simplify the expression further.

The law of excluded middle states that: \[p \lor \sim p \equiv T\] Applying the law of excluded middle, we have: \[\sim p \land (\sim q \lor q) \equiv \sim p \land T\]

5. Apply the identity law to arrive at the simplest form.

The identity law states that: \[p \land T \equiv p\] So, \[\sim p \land T \equiv \sim p\] So the simplified expression is not: (a) q - as \(\sim p \neq q\), (b) p - as \(\sim p \neq p\), (d) ¬q - as \(\sim p \neq \sim q\) But it is: (c) ¬p - as \(\sim p \equiv \sim p\) The final solution is (c) ¬p

Key concepts

Logic Symbols

When diving into the fascinating world of logical expressions, the first thing you'll encounter is a set of symbols—these are the ABCs of logic that we use to represent complex ideas in a clear, concise way.
Let's talk about a few essentials: The symbol eq is 'not', which denotes negation, flipping the truth value of a statement. Then there's ∧, called 'and', signifying a conjunction where both conditions must be true for the entire expression to be true. The ∨ symbol represents 'or', known as a disjunction, where only one of the conditions needs to be true. The equality symbol ≡ is used for logical equivalence, stating that two expressions always share the same truth value. Understanding how to read and write these symbols is crucial because they are the building blocks of logical reasoning and mathematical proofs.
A well-informed student in logic symbols can accurately interpret and construct logical statements, a skill invaluable in mathematics, computer science, and philosophy.

Distributive Law

Imagine you're at a store, dividing your shopping items between two bags. Just as you might distribute the items evenly, the distributive law allows us to evenly 'distribute' logical operators over others. In a logical context, this law tells us how conjunctions and disjunctions interact.
Specifically, the distributive law comes in two variants:

1. eq A ∧ (B ∨ C) ≡ (A ∧ B) ∨ (A ∧ C)
2. A ∨ (B ∧ C) ≡ (A ∨ B) ∧ (A ∨ C)

This essentially means that if you have a statement like 'I will bring an umbrella or (a hat and sunglasses)', it's the same as saying 'I will bring an umbrella or a hat, and an umbrella or sunglasses'. It's a simple yet powerful rule that prevents logical expressions from becoming tangled messes, much like keeping your shopping bags organized.

Law of Excluded Middle

The law of excluded middle is like having to decide if you'll go out or stay in—it can only be one or the other, not both or neither. This law is a cornerstone of classical logic, stating that any proposition is either true or its negation is true. In symbol terms: P ∨ ¬P ≡ T (where T stands for 'true').
This principle asserts that there is no middle ground, no 'half-true'. For any statement P, there can't be any ambiguity; P is either true, or not-P is true. There's no room for maybe. This law is particularly handy when we simplify complex logical expressions. It's what guarantees that, no matter what, the expression ¬Q ∨ Q will always evaluate to true, providing a solid foundation when we're streamlining logical formulas.

Identity Law

The identity law in logic is much like your ID card; it confirms who you are without any doubt. In logical terms, this law helps us to confirm the true nature of a statement without changing its essence.
There are two aspects to the identity law:

1. P ∧ T ≡ P
2. P ∨ F ≡ P

Here T is the truth constant, meaning that 'P and true' will always just be P, and F is the false constant, meaning 'P or false' is also just P. It's a way of stripping down the expression to its core identity, removing any true or false additions that don't actually change the meaning. Think of it as clearing away the clutter to reveal the truth of the statement.