Logical Inference: First Order Logic Quiz

1. What does '→' represent in first-order logic?

XOR

AND

NOT

IMPLIES

In first-order logic, the symbol '→' represents the logical connective for implication or conditional. It is read as "implies" or "if-then." The expression P→Q asserts that if proposition P is true, then proposition Q must also be true. If P is false, or if P is true and Q is false, the entire statement is considered true.

Explanation

In first-order logic, the symbol '→' represents the logical connective for implication or conditional. It is read as "implies" or "if-then." The expression P→Q asserts that if proposition P is true, then proposition Q must also be true. If P is false, or if P is true and Q is false, the entire statement is considered true.

2. In first-order logic, '∃' represents which type of quantifier?

FOR SOME

NOT

IF THEN

AND

In first-order logic, the symbol '∃' represents the existential quantifier, which is equivalent to "FOR SOME." The existential quantifier (∃) is used to express that there exists at least one instance of a variable for which a given statement is true.

Explanation

In first-order logic, the symbol '∃' represents the existential quantifier, which is equivalent to "FOR SOME." The existential quantifier (∃) is used to express that there exists at least one instance of a variable for which a given statement is true.

3. Which logical relationship is represented by the symbol '∨' in first-order logic?

NOR

NOT

XOR

OR

In first-order logic, the symbol '∨' represents the logical connective for disjunction or logical OR. The expression P ∨ Q is true if at least one of the propositions P or Q is true, and it is false only when both P and Q are false.

Explanation

In first-order logic, the symbol '∨' represents the logical connective for disjunction or logical OR. The expression P ∨ Q is true if at least one of the propositions P or Q is true, and it is false only when both P and Q are false.

4. Which of the following is a quantifier in first-order logic?

NOT

AND

FOR ALL

OR

In first-order logic, "FOR ALL" (∀) is a universal quantifier that indicates a statement is true for every possible instance of a variable. It is used to express that a particular property or relation holds for all members of a given set.

Explanation

In first-order logic, "FOR ALL" (∀) is a universal quantifier that indicates a statement is true for every possible instance of a variable. It is used to express that a particular property or relation holds for all members of a given set.

5. What is the negation of '∃x P(x)'?

∃x NOT P(x)

∀x ¬P(x)

∀x NOT P(x)

∃x P(x)

The correct negation of '∃x P(x)' is: ∀x ¬P(x)

This is read as "For all x, P(x) is not true" or "It is not the case that there exists an x for which P(x) is true." It asserts that for every x, the property P(x) is not true.

Explanation

The correct negation of '∃x P(x)' is: ∀x ¬P(x)

This is read as "For all x, P(x) is not true" or "It is not the case that there exists an x for which P(x) is true." It asserts that for every x, the property P(x) is not true.

6. What does '↔' stand for in first order logic?

AND

IFF

XOR

IMPLIES

In first-order logic, the symbol '↔' represents the logical connective for biconditional or "if and only if" (IFF). It is read as "if and only if" and is true when both propositions on either side of the symbol have the same truth value (both true or both false) and false otherwise. The expression ( P↔Q ) asserts that P is true if and only if Q is true, and vice versa.

Explanation

In first-order logic, the symbol '↔' represents the logical connective for biconditional or "if and only if" (IFF). It is read as "if and only if" and is true when both propositions on either side of the symbol have the same truth value (both true or both false) and false otherwise. The expression ( P↔Q ) asserts that P is true if and only if Q is true, and vice versa.

7. Which logical relationship is denoted by '⊕' in first order logic?

XOR

AND

NAND

NOR

In first-order logic, the symbol '⊕' represents the logical connective for XOR (exclusive or). The expression (P ⊕ Q) is true when either P is true or Q is true, but not both. It is false when both P and Q have the same truth value (both true or both false). The symbol '⊕' denotes the exclusive or relationship in logical expressions.

Explanation

In first-order logic, the symbol '⊕' represents the logical connective for XOR (exclusive or). The expression (P ⊕ Q) is true when either P is true or Q is true, but not both. It is false when both P and Q have the same truth value (both true or both false). The symbol '⊕' denotes the exclusive or relationship in logical expressions.

8. What is the negation of '∀x P(x)'?

∀x NOT P(x)

∃x ¬P(x)

∃x NOT P(x)

∀x P(x)

The negation of the statement '∀x P(x)' (which reads as "For all x, P(x)") is represented as '∃x ¬P(x)' (which reads as "There exists an x for which P(x) is not true"). In other words, it asserts that it is not the case that every x satisfies the property P; there is at least one x for which P(x) is false.

Explanation

The negation of the statement '∀x P(x)' (which reads as "For all x, P(x)") is represented as '∃x ¬P(x)' (which reads as "There exists an x for which P(x) is not true"). In other words, it asserts that it is not the case that every x satisfies the property P; there is at least one x for which P(x) is false.

9. Which quantifier is used to express uniqueness in first order logic?

∀

∃!

∀∃

∃∀

The quantifier used to express uniqueness in first-order logic is: ∃!

The symbol ∃! is read as "there exists a unique" and is used to assert that there is exactly one element satisfying a given property. It combines the existential quantifier (∃) with the uniqueness assertion (!). For example, ∃!x P(x) would be read as "There exists a unique x such that P(x) is true."

Explanation

The quantifier used to express uniqueness in first-order logic is: ∃!

The symbol ∃! is read as "there exists a unique" and is used to assert that there is exactly one element satisfying a given property. It combines the existential quantifier (∃) with the uniqueness assertion (!). For example, ∃!x P(x) would be read as "There exists a unique x such that P(x) is true."

10. Which of the following is a predicate in first-order logic?

OR

AND

∀

IS EVEN

In first-order logic, a predicate is a statement that contains variables and becomes a proposition when specific values are substituted for these variables. "IS EVEN" is an example of a predicate, as it involves a property (evenness) that can be applied to a variable.

Explanation

In first-order logic, a predicate is a statement that contains variables and becomes a proposition when specific values are substituted for these variables. "IS EVEN" is an example of a predicate, as it involves a property (evenness) that can be applied to a variable.