Homework Help: Questions and Answers: Decide whether or not the following are statements In the case of a statement say if it is true or false if possible:
- Every real number is a positive integer.
- Every positive integer is a real number.
- If x and y are real numbers and x=y, then 5x=5y.
- x⋅x^(-1)=1
Answer
Let’s analyze each of the given sentences (options) step by step to determine whether they are statements. If they are statements, we’ll also determine if they are true or false.
1. Every real number is a positive integer.
- A real number can be any number on the continuous number line, including fractions, decimals, negatives, zero, and positive integers. A positive integer is a whole number greater than zero (e.g., 1, 2, 3, …).
- This is a statement because it is a declarative sentence that is either true or false.
- Truth Value: The statement is false. Not all real numbers are positive integers. For example, -1, 0, and 1/2 are real numbers but not positive integers.
2. Every positive integer is a real number.
- As mentioned earlier, positive integers are whole numbers greater than zero. Real numbers include all integers (positive, negative, and zero) as well as fractions and decimals.
- This is a statement because it is a declarative sentence that is either true or false.
- Truth Value: The statement is true. Every positive integer (like 1, 2, 3, …) is indeed a real number.
3. If x and y are real numbers and x=y, then 5x=5y.
- If x=y, then multiplying both sides of the equation by the same number (in this case, 5) will result in the equation 5x=5y.
- This is a statement because it is a declarative sentence that is either true or false.
- Truth Value: The statement is true. The equation holds for any real numbers x and y.
4. x⋅x^(-1)=1
- Here, x⋅x^(-1)=1 represents the multiplicative inverse of x, which means x⋅x^(-1)=1 for any non-zero x.
- This is a statement because it is a declarative sentence that is either true or false.
- Truth Value: The statement is true for any non-zero real number x. However, it is undefined for x=0, as the multiplicative inverse of 0 does not exist.
Conclusion
- Every real number is a positive integer. – False.
- Every positive integer is a real number. – True.
- If x and y are real numbers and x=y, then 5x=5y – True.
- x⋅x^(-1)=1 – True (for x≠0).
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