Math Problem Statement

Let π‘ƒπ‘Ÿπ‘–π‘šπ‘’(𝑛) be the predicate β€œπ‘› 𝑖𝑠 π‘Ž π‘π‘Ÿπ‘–π‘šπ‘’ π‘›π‘’π‘šπ‘π‘’π‘Ÿ" and 𝐼𝑛 (𝑛,π‘Ž,𝑏) be the predicate π‘Žβ‰€π‘›β‰€π‘. Where 𝑛, π‘Ž and 𝑏 are in the domain of all integers. For each of the following statements, indicate if they are True, False or Neither. (For the tutorial submission answer at least 5 of the below (a-j) and provide a 1-line explanation for each). Predicate True / False / Neither Explanation Example: π‘ƒπ‘Ÿπ‘–π‘šπ‘’(12)βˆ§πΌπ‘›(12,5,10) False π‘ƒπ‘Ÿπ‘–π‘šπ‘’(12)=πΉπ‘Žπ‘™π‘ π‘’ πΉπ‘Žπ‘™π‘ π‘’βˆ§π‘Žπ‘›π‘¦π‘‘β„Žπ‘–π‘›π‘”=πΉπ‘Žπ‘™π‘ π‘’ a. Β¬π‘ƒπ‘Ÿπ‘–π‘šπ‘’(10)βˆ¨πΌπ‘›(10,5,20) b. βˆƒπ‘›π‘ƒπ‘Ÿπ‘–π‘šπ‘’(𝑛) c. βˆƒπ‘›Β¬π‘ƒπ‘Ÿπ‘–π‘šπ‘’(𝑛) d. βˆ€π‘›π‘ƒπ‘Ÿπ‘–π‘šπ‘’(𝑛) e. βˆ€π‘›Β¬π‘ƒπ‘Ÿπ‘–π‘šπ‘’(𝑛) f. Β¬βˆ€π‘›π‘ƒπ‘Ÿπ‘–π‘šπ‘’(𝑛) g. βˆ€π‘›(𝐼𝑛(𝑛,1,3)β†’π‘ƒπ‘Ÿπ‘–π‘šπ‘’(𝑛)) h. βˆ€π‘›(𝐼𝑛(𝑛,8,10)β†’π‘ƒπ‘Ÿπ‘–π‘šπ‘’(𝑛)) i. βˆ€π‘›(𝐼𝑛(𝑛,π‘Ž,𝑏)β†’Β¬π‘ƒπ‘Ÿπ‘–π‘šπ‘’(𝑛)) Where π‘Ž and 𝑏 are integer smaller than 10 j. βˆƒπ‘›π‘ƒπ‘Ÿπ‘–π‘šπ‘’(𝑛)→𝐼𝑛(𝑛,30,40)

Let Prime(n) be the predicate β€œn is a prime number” and In(n,a,b) be the predicate a ≀ n ≀ b. Where n, a and b are in the domain of all integers. For each of the following statements, indicate if they are True, False or Neither.

Solution

Let's analyze and provide explanations for at least 5 of the predicates from the list:

a. ¬Prime(10)∨In(10,5,20)\neg Prime(10) \lor In(10,5,20)

  • True/False/Neither: True
  • Explanation: Prime(10)Prime(10) is False because 10 is not a prime number, but In(10,5,20)In(10,5,20) is True because 5≀10≀205 \leq 10 \leq 20. Thus, the disjunction (OR) is True.

b. βˆƒnPrime(n)\exists n Prime(n)

  • True/False/Neither: True
  • Explanation: There exists at least one prime number, so this statement is True. For example, Prime(2)Prime(2) is True.

c. βˆƒnΒ¬Prime(n)\exists n \neg Prime(n)

  • True/False/Neither: True
  • Explanation: There exists at least one non-prime number, such as 1, 4, or 10. Therefore, the statement is True.

d. βˆ€nPrime(n)\forall n Prime(n)

  • True/False/Neither: False
  • Explanation: Not all numbers are prime. For instance, 1, 4, and 10 are not prime, making this universally quantified statement False.

g. βˆ€n(In(n,1,3)β†’Prime(n))\forall n (In(n,1,3) \to Prime(n))

  • True/False/Neither: False
  • Explanation: The range In(n,1,3)In(n,1,3) includes n=1,2,3n = 1, 2, 3. While 2 and 3 are prime, 1 is not a prime number, making this implication False.

Let me know if you'd like further details or have any questions! Here are 5 follow-up questions to explore the topic further:

  1. How would the truth value of part hh change if the interval were In(n,7,10)In(n,7,10)?
  2. Can you give an example of a universally quantified statement involving composite numbers?
  3. Why is Prime(1)Prime(1) not true, even though 1 is sometimes considered special in number theory?
  4. Could there be a situation where both the existential quantifier and the universal quantifier hold in the same interval?
  5. How does the definition of a prime number affect other intervals like In(n,20,30)In(n,20,30)?

Tip: When dealing with quantifiers like βˆ€\forall (for all) or βˆƒ\exists (there exists), it's crucial to find counterexamples for falsehood or confirmations for truth!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Prime Numbers
Logical Operators (AND, OR, NOT)
Quantifiers (Existential βˆƒ, Universal βˆ€)
Number Theory
Intervals in Mathematics

Formulas

-

Theorems

Prime Number Definition
Logical Conjunction and Disjunction

Suitable Grade Level

Grades 10-12

Related Recommendation

Logical Translation of Relative Primeness and Divisibility

Expressing Integer Properties Using Logical Symbols

Logical Reasoning with Prime Numbers and Divisibility by 5

Analyzing Predicate Logic Statement with Integer Sets

Truth Table for '5 is a Prime and Composite Number'