Negated Conjunction in Predicate Logic: P ⇔ (∀x) (x ∧ ¬y)

twoflower · 2007-01-12T13:11:20+00:00

Homework Statement p \leftrightarrow \left( \forall x \right)\left( x \wedge \neg y \right) Homework Equations The Attempt at a Solution

Thread starter twoflower
Start date Jan 12, 2007

In summary, a negated conjunction in predicate logic is a logical statement that combines two propositions using "∧" and negates one of the propositions using "¬". It differs from regular conjunction in that it includes the negation of one of the propositions. The symbol "⇔" represents the biconditional operator, indicating that the two propositions are equivalent. The statement P ⇔ (∀x) (x ∧ ¬y) is read as "P if and only if for all x, x and not y are true." An example of a negated conjunction is "The cat is black ∧ ¬The dog is brown."

Jan 12, 2007

twoflower

Homework Statement

[tex]p \leftrightarrow \left( \forall x \right)\left( x \wedge \neg y \right)[/tex]

Homework Equations

The Attempt at a Solution

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Jan 12, 2007

matt grime

Science Advisor

Homework Helper

The words next to 1. really aren't just there for decoration.

1. What is a negated conjunction in predicate logic?

A negated conjunction in predicate logic is a logical statement that combines two propositions using the symbol "∧" (and) and negates one of the propositions using the symbol "¬" (not). It is represented as P ∧ ¬Q, where P and Q are propositions.

2. How is negated conjunction different from regular conjunction?

Negated conjunction differs from regular conjunction in that it includes the negation of one of the propositions. In regular conjunction, both propositions must be true for the statement to be true, while in negated conjunction, one of the propositions must be true and the other must be false for the statement to be true.

3. What does the symbol "⇔" mean in the statement P ⇔ (∀x) (x ∧ ¬y)?

The symbol "⇔" represents the biconditional or "if and only if" operator. In this statement, it indicates that the two propositions (P and (∀x) (x ∧ ¬y)) are equivalent and have the same truth value.

4. How is the statement P ⇔ (∀x) (x ∧ ¬y) read?

The statement P ⇔ (∀x) (x ∧ ¬y) is read as "P if and only if for all x, x and not y are true." This means that P is true if and only if all instances of x being true and y being false are true.

5. Can you provide an example of a negated conjunction in predicate logic?

One example of a negated conjunction in predicate logic is "The cat is black ∧ ¬The dog is brown." This statement combines two propositions, "The cat is black" and "The dog is brown," and negates the second proposition. Therefore, the statement is true if and only if the cat is black and the dog is not brown.