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Quantifiers are symbols used in logic to express the extent of applicability of a predicate to a subject. There are two main types of quantifiers: universal and existential.
Universal quantifiers (\(\forall\)) denote that a statement applies to all elements within a certain domain. For example, \(\forall x (P(x))\) means 'For all x, P(x) is true.'
Existential quantifiers (\(\there exists\)) indicate that there is at least one element in the domain for which the statement is true. For example, \(\there exists x (P(x))\) means 'There exists at least one x such that P(x) is true.'
In the given exercise:

• Statement (a) 'No professors are ignorant' uses the universal quantifier \(\forall\) to signify that for every professor, they are not ignorant.
• Statement (b) 'All ignorant people are vain' also uses the universal quantifier to state that for all ignorant people, they are vain.
• Statement (c) 'No professors are vain' is expressed similarly using the universal quantifier to declare that no professors possess vanity.

Logical connectives are used to combine or modify statements and are crucial for constructing complex logical expressions.
The main logical connectives are:

• AND (\(\wedge\)): Combines two statements and is true only if both statements are true.
  Example: \(P(x) \wedge Q(x)\) 'x is a professor and x is ignorant.'
• OR (\(\vee\)): Combines two statements and is true if at least one statement is true.
  Example: \(P(x) \vee Q(x)\) 'x is a professor or x is ignorant.'
• NOT (\(eg\)): Negates a statement, flipping its truth value.
  Example: \(eg Q(x)\) 'x is not ignorant.'
• IMPLICATION (\(\rightarrow\)): Indicates that one statement implies another. It's true except when the first statement is true and the second is false.
  Example: \(P(x) \rightarrow eg Q(x)\) 'If x is a professor, then x is not ignorant.'

In our exercise:

• Statement (a) 'No professors are ignorant' uses \(\rightarrow\) to say 'If x is a professor, then x is not ignorant.'
• Statement (b) 'All ignorant people are vain' also uses \(\rightarrow\) to imply 'If x is ignorant, then x is vain.'
• Statement (c) 'No professors are vain' uses \(\rightarrow\) to declare 'If x is a professor, then x is not vain.'

Contrapositive is a logically equivalent form of an implication, which states that if the negation of the conclusion implies the negation of the premise, then the original implication holds.
In other words, for a statement of the form \(P(x) \rightarrow Q(x)\), its contrapositive is \(eg Q(x) \rightarrow eg P(x)\). If one is true, the other one is also true.
In our exercise, statement (b) 'All ignorant people are vain' or \(\forall x (Q(x) \rightarrow R(x))\) has a contrapositive form: \(\forall x (eg R(x) \rightarrow eg Q(x))\). This helps in proving statement (c).
To determine if statement (c) 'No professors are vain' follows from (a) and (b), we use the contrapositive of (b).

• From (a): \(\forall x (P(x) \rightarrow eg Q(x))\)
• From (b): \(\forall x (Q(x) \rightarrow R(x))\) Contrapositive: \(\forall x (eg R(x) \rightarrow eg Q(x))\)
• Combining these, we deduce: \(\forall x (P(x) \rightarrow eg R(x))\)

This shows that (c) follows from (a) and (b) using contrapositive reasoning.

Implication (\(\rightarrow\)) is a logical connective that indicates a conditional relationship between two statements. An implication statement \(P(x) \rightarrow Q(x)\) asserts that if the premise \(P(x)\) is true, then the conclusion \(Q(x)\) must also be true.
Key characteristics of implications:

• The statement \(P(x) \rightarrow Q(x)\) is false only when \(P(x)\) is true and \(Q(x)\) is false.
• If \(P(x)\) is false, \(Q(x)\) can be either true or false, making the implication true.

In our exercise:

• Statement (a) 'No professors are ignorant' translates to \(\forall x (P(x) \rightarrow eg Q(x))\), indicating that if x is a professor, then x is not ignorant.
• Statement (b) 'All ignorant people are vain' translates to \(\forall x (Q(x) \rightarrow R(x))\), indicating that if x is ignorant, then x is vain.
• Statement (c) 'No professors are vain' translates to \(\forall x (P(x) \rightarrow eg R(x))\), indicating that if x is a professor, then x is not vain.

The exercise involves determining if the statement in (c) can be logically derived using implications from statements (a) and (b). By recognizing that the contrapositive of statement (b) and statement (a) lead to the conclusion in (c), we establish the logical connection.