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Express each of these statements using predicates and quantifiers. a) A passenger on an airline qualifies as an elite flyer if the passenger flies more than 25,000 miles in a year or takes more than 25 flights during that year. b) A man qualifies for the marathon if his best previous time is less than 3 hours and a woman qualifies for the marathon if her best previous time is less than 3.5 hours. c) A student must take at least 60 course hours, or at least 45 course hours and write a master鈥檚 thesis, and receive a grade no lower than a B in all required courses, to receive a master鈥檚 degree. d) There is a student who has taken more than 21 credit hours in a semester and received all A鈥檚.

Step by step solution

01

Define predicates and variables for statement a

Let P(x) represent 'x is a passenger on an airline', M(x) represent 'x flies more than 25,000 miles in a year', and F(x) represent 'x takes more than 25 flights during that year'. The statement can be expressed as: \[\forall x(P(x) \rightarrow (M(x) \lor F(x)))\]

02

Define predicates and variables for statement b

Let M(x) represent 'x is a man', W(x) represent 'x is a woman', and T(x, y) represent 'x's best previous time is less than y'. The statements can be expressed as:For men: \[\forall x(M(x) \rightarrow T(x, 3))\]For women: \[\forall x(W(x) \rightarrow T(x, 3.5))\]

03

Define predicates and variables for statement c

Let S(x) represent 'x is a student', C(x, y) represent 'x takes at least y course hours', T(x) represent 'x writes a master鈥檚 thesis', and G(x, B) represent 'x receives a grade no lower than a B in all required courses'. The statement can be expressed as:\[\forall x(S(x) \rightarrow ((C(x, 60) \lor (C(x, 45) \land T(x))) \land G(x, B)))\]

04

Define predicates and variables for statement d

Let S(x) represent 'x is a student', C(x, y) represent 'x has taken more than y credit hours in a semester', and A(x) represent 'x received all A鈥檚'. The statement can be expressed as:\[\exists x(S(x) \land C(x, 21) \land A(x))\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantifiers

Quantifiers are essential components in predicate logic. They help us specify the scope of a statement concerning the elements of a set. There are two main types of quantifiers: universal and existential.

• Universal Quantifier (\(\forall\)): Indicates that the statement is true for all elements of the domain. For example, \(\forall x P(x)\) means 'P(x) is true for all x.'
• Existential Quantifier (\(\forall\)): Indicates that there is at least one element in the domain for which the statement is true. For example, \(\forall x P(x)\) means 'There exists an x for which P(x) is true.'

In our exercise, we saw the use of both quantifiers.
For instance, \(\forall x P(x) \rightarrow (M(x) \forall F(x))\) states that for all passengers, if they qualify as elite flyers, they must either fly more than 25,000 miles or take more than 25 flights in a year.

Predicates

Predicates are statements that contain variables and become propositions when specific values are substituted for those variables. They are usually denoted by uppercase letters like P, Q, R, etc.
For example, P(x) could mean 'x is a passenger on an airline.'
Our exercise included several predicates, such as:

• \(P(x)\): x is a passenger on an airline
• \(M(x)\): x flies more than 25,000 miles in a year
• \(F(x)\): x takes more than 25 flights during that year

By combining these predicates with quantifiers and logical connectors, we can form logical expressions that represent complex statements.

Logical Expressions

Logical expressions in predicate logic combine predicates with quantifiers and logical connectives such as AND (\(\forall\)), OR (\(\forall\)), and NOT (\(\forall\)).
Consider the following examples from our exercise:
\(\forall x(M(x) \forall T(3))\) indicates that for all men, their best previous time is less than 3 hours.
Another example is \(\forall x(S(x) \rightarrow ((C(x, 60) \forall (C(x, 45) \rightarrow T(x))) \forall G(x, B)))\), which states that to receive a master's degree, a student must take at least 60 course hours or 45 course hours and write a master's thesis and also receive no grade lower than a B in all required courses.

Mathematical Reasoning

Mathematical reasoning is the process of using logic to solve problems and prove statements. It allows us to structure arguments and statements systematically.
This exercise demonstrates mathematical reasoning by breaking down complex conditions into simpler components (predicates) and structuring them logically. For instance, the statement about qualifiers for the marathon is divided into:

• \(\forall x(M(x) \rightarrow T(x, 3))\): A man qualifies if his best previous time is less than 3 hours
• \(\forall x(W(x) \rightarrow T(x, 3.5))\): A woman qualifies if her best previous time is less than 3.5 hours

By defining each condition clearly and using logical expressions to combine them, we apply mathematical reasoning to express and solve the problem accurately.