Question

write each sentence as a conditional statement. 4. robert oppenheimer spoke eight languages. 5. purple and yellow are complementary colors. determine if the conditional is true or false. if it is false, find a counterexample. 6. obtuse angles have measures greater than 90. 7. if n is a prime number, then it is divisible by 2. write the two statements that form each biconditional. 8. the product of two numbers is a perfect square if and only if the two numbers are identical. 9. a number is a terminating decimal if and only if it contains a finite number of digits. 10. a figure is three - dimensional if and only if it has length, width, and height.

Explanation:

Step1: Rewrite as conditional statements

For 4:

If a person is Robert Oppenheimer, then the person spoke eight languages.

For 5:

If two colors are purple and yellow, then they are complementary colors.

Step2: Determine truth - value of conditionals

For 6:

The conditional "Obtuse angles have measures greater than 90" is true by the definition of obtuse angles (obtuse angles are defined as angles with measures between 90° and 180°).

For 7:

The conditional "If n is a prime number, then it is divisible by 2" is false. Counter - example: 3 is a prime number, but 3 is not divisible by 2.

Step3: Write biconditional as two statements

For 8:

Statement 1: If the product of two numbers is a perfect square, then the two numbers are identical.
Statement 2: If two numbers are identical, then the product of the two numbers is a perfect square.

For 9:

Statement 1: If a number is a terminating decimal, then it contains a finite number of digits.
Statement 2: If a number contains a finite number of digits, then it is a terminating decimal.

For 10:

Statement 1: If a figure is three - dimensional, then it has length, width, and height.
Statement 2: If a figure has length, width, and height, then it is three - dimensional.

Answer:

1. If a person is Robert Oppenheimer, then the person spoke eight languages.
2. If two colors are purple and yellow, then they are complementary colors.
3. True
4. False, counter - example: 3
5. Statement 1: If the product of two numbers is a perfect square, then the two numbers are identical. Statement 2: If two numbers are identical, then the product of the two numbers is a perfect square.
6. Statement 1: If a number is a terminating decimal, then it contains a finite number of digits. Statement 2: If a number contains a finite number of digits, then it is a terminating decimal.
7. Statement 1: If a figure is three - dimensional, then it has length, width, and height. Statement 2: If a figure has length, width, and height, then it is three - dimensional.