Classify each of the following statements as true, false or not a valid mathematical statement and explain your answers. a. Suppose this statement is false. b. The product of five prime numbers is prime
Classify each of the following statements as true, false or not a valid mathematical statement and explain your answers.
a. Suppose this statement is false.
b. The product of five prime numbers is prime
Rewrite each of the following statements so that it is clear that each is an implication a. n is not an even integer for any integer n.
n+1
b. n + 3 > 2 for every natural number n.
The correct answer and explanation is :
Part 1: Classifying the statements
a. Suppose this statement is false.
This statement is not a valid mathematical statement. It is a conditional statement that lacks a well-defined proposition or logical structure. In mathematical reasoning, every statement must have a clear truth value (true or false), and in this case, the statement is self-referential but doesn’t describe any particular fact. Without specifying what “this statement” refers to, it is unclear and therefore not a valid mathematical statement.
b. The product of five prime numbers is prime.
This statement is false. The product of multiple prime numbers greater than 1 is always composite, not prime. A prime number is a number that has exactly two distinct positive divisors: 1 and itself. When multiplying five primes together, the result will have more than two divisors (for example, 2 × 3 × 5 × 7 × 11 = 2310), and thus it will be composite. Therefore, the statement is false.
Part 2: Rewriting statements as implications
a. n is not an even integer for any integer n.
To rewrite this as an implication, we need to express it in the form “If [condition], then [result].” The statement means that if (n) is an integer, then (n) is not even. This can be rewritten as:
- If n is an integer, then n is not an even integer.
This statement says that for any integer (n), it cannot be an even number. However, note that this is actually false because integers such as 2, 4, 6, etc., are even. So, the correct way of expressing this in implication form would be logically incorrect in the context of even integers.
b. n + 3 > 2 for every natural number n.
To rewrite this as an implication, we again express it in the “If [condition], then [result]” form. The statement means that for any natural number (n), adding 3 to (n) will always result in a value greater than 2. This can be rewritten as:
- If n is a natural number, then n + 3 > 2.
This is a valid mathematical implication. The statement is true because for any natural number (n), which is at least 1, adding 3 will always result in a number greater than 2. Therefore, this implication holds true for all natural numbers.
Conclusion:
- Part 1:
- “Suppose this statement is false” is not valid because it’s incomplete.
- “The product of five prime numbers is prime” is false because the product of prime numbers greater than 1 is always composite.
- Part 2:
- The first statement (“n is not an even integer for any integer n”) is misleading and false, but when rewritten as an implication, it’s expressed as: “If n is an integer, then n is not an even integer.”
- The second statement (“n + 3 > 2 for every natural number n”) is a true statement and can be rewritten as: “If n is a natural number, then n + 3 > 2.”