Consider the sentence below. There is a country that borders both India and Pakistan. Which of the following logical expressions express the above sentence correctly when the predicate Country(x) represents that x is a country and Borders(x, y) represents that the countries x and y share the border?
∃c Country(c) ∧ Border (c, India) ∧ Border (c, Pakistan)
Understanding the Logic Translation Problem
The question asks us to translate a natural language sentence into a formal logical expression using given predicates. The sentence is: "There is a country that borders both India and Pakistan."
We are provided with two predicates:
- Country(x): Represents that x is a country.
- Borders(x, y): Represents that the countries x and y share a border.
Our goal is to find the logical expression among the options that correctly captures the meaning of the given sentence.
Breaking Down the Sentence for Translation
Let's break down the sentence "There is a country that borders both India and Pakistan" into its core logical components:
- Existence: The phrase "There is a country" implies the existence of at least one entity with certain properties. In first-order logic, this is represented by the existential quantifier, ∃. Let's use the variable 'c' to represent this entity. So, we start with ∃c.
- Condition 1: The entity 'c' must be a country. This is represented by the predicate Country(c).
- Condition 2: This country 'c' must border India. Using the given predicate, this is represented by Borders(c, India).
- Condition 3: This country 'c' must also border Pakistan. Using the given predicate, this is represented by Borders(c, Pakistan).
- Combining Conditions: The sentence requires that the country 'c' satisfies ALL these conditions simultaneously (it is a country AND it borders India AND it borders Pakistan). In logic, the "AND" connective is represented by & (& in HTML).
Constructing the Logical Expression
Putting the components together, the sentence "There is a country 'c' such that 'c' is a country AND 'c' borders India AND 'c' borders Pakistan" translates to:
∃c (Country(c) & Borders(c, India) & Borders(c, Pakistan))
Let's compare this derived expression with the given options.
Analyzing the Given Logical Expression Options
We will now evaluate each option based on our understanding of the sentence and the correct translation rules of first-order logic.
| Option | Logical Expression | Analysis |
|---|---|---|
| 1 | ∃c Country(c) & Border (c, India) & Border (c, Pakistan) | This expression states, "There exists an entity 'c' such that 'c' is a country AND 'c' borders India AND 'c' borders Pakistan." This exactly matches the meaning of the original sentence. The variable 'c' is bound by the existential quantifier throughout the expression. |
| 2 | ∃c Country(c) ⇒ [Border (c, India) & Border (c, Pakistan)] | This translates to "There exists an entity 'c' such that IF 'c' is a country, THEN 'c' borders India and Pakistan." This does NOT capture the original sentence. The original sentence asserts the existence of such a country. This implication would be true for *any* 'c' that is not a country, regardless of whether it borders India or Pakistan. It doesn't guarantee the existence of a country with the specified properties. |
| 3 | [∃c Country(c)] ⇒ [Border (c, India) & Border (c, Pakistan)] | This translates to "IF there exists at least one country in the world, THEN India borders Pakistan." This is completely different from the original sentence. It makes a conditional statement about the general existence of countries and a specific border relationship between India and Pakistan, not about a single country bordering both. Also, the predicate usage Border(c, India) & Border(c, Pakistan) within the consequent is unclear in this context; it should refer to a specific entity or be Border(India, Pakistan) if that's the intended meaning (which it isn't based on the original sentence structure). |
| 4 | ∃c Border (Country(c), India & Pakistan) | This expression is syntactically incorrect in standard first-order logic. Predicates like Borders take terms (like 'c', 'India', 'Pakistan') as arguments. Country(c) is a predicate that evaluates to true or false, not a term representing the country itself. Similarly, India & Pakistan combines terms with a logical connective (&), which is not how arguments for predicates are formed. |
Based on the analysis, only Option 1 correctly translates the sentence "There is a country that borders both India and Pakistan" into a formal logical expression using the given predicates and the existential quantifier.
Conclusion: Correct Logical Expression
The correct logical expression that accurately represents the sentence "There is a country that borders both India and Pakistan" is the one stating the existence of an entity 'c' which is a country AND borders India AND borders Pakistan. This corresponds directly to Option 1.
The correct option is:
∃c Country(c) & Border (c, India) & Border (c, Pakistan)
Revision Table: Key Logic Concepts
| Concept | Symbol | Meaning | Relevance to Problem |
|---|---|---|---|
| Existential Quantifier | ∃ | "There exists" or "For some" | Used to express "There is a country..." |
| Conjunction | & (or ∧) | "AND" | Used to combine the conditions: being a country AND bordering India AND bordering Pakistan. |
| Implication | ⇒ (or →) | "IF... THEN..." | Appears in incorrect options; represents a conditional relationship, not the assertion of existence with properties. |
| Predicate | e.g., Country(x), Borders(x,y) | Represents a property or relation | Given in the problem to define the terms used in the logical expression. |
Additional Information: First-Order Logic Basics
First-Order Logic (FOL), also known as Predicate Logic, is a formal system used to represent statements about objects and their properties or relations. It extends propositional logic by introducing quantifiers (∀ for "for all" and ∃ for "there exists") and predicates.
- Terms: Represent objects or entities (e.g., 'c', 'India', 'Pakistan').
- Predicates: Represent properties of objects or relations between objects (e.g., Country(x), Borders(x, y)).
- Atomic Formulas: Formed by applying a predicate to appropriate terms (e.g., Country(India), Borders(India, Pakistan)).
- Logical Connectives: Used to combine atomic formulas (e.g., & (AND), ∨ (OR), ¬ (NOT), ⇒ (IMPLIES), ⇔ (IF AND ONLY IF)).
- Quantifiers: Used to express the scope of statements over a domain of discourse. ∃ asserts that something is true for at least one object, while ∀ asserts that something is true for every object.
Translating natural language sentences into FOL requires carefully identifying the objects, properties, relations, and the scope of any claims about existence or universality.
Important Questions from Multiplexer
The device which changes from serial data to parallel data is:
Consider a logic gate circuit. with 8 input lines (D 0, D 1..... D 7) and 3 output lines (A 0, A 1, A 2) specified by following operations
A 2= D 4+ D 5+ D 6+ D 7
A 1= D 2+ D 3+ D 6+ D 7
A 0= D 1+ D 3+ D 5+ D 0
Where + indicates logical OR operation. This circuit is
Which of the following statements are true?
A. Frequency division multiplexing technique can be handled by digital circuits.
B. Time division multiplexing technique can be handled by analog circuits
C. Wavelength division multiplexing technique is used with optical fiber for combining two signals.
D. Frequency division multiplexing technique can be applied when the bandwidth of a link is greater than the bandwidth of the signals to be transmitted.
Choose the correct answer from the options given below :
A digital computer has a common bus system for 8 registers 16 bits each. How many multiplexers are required to implement common bus? What size of multiplexers is required?
In a multiplexer, the number of input lines is ‘n’ and the number of select lines is ‘m’. Which of the following is correct?
