Determine whether the following sets are closed under addition, subtraction, maldiplication, and division. 1. (-1,0,1) 2. 0,8 3. x^2,1 4. 0,x 5. -x^3,1,x^3 (-x,1,x+1) 7. (-1,1) 8. (-1,0,x] 9 (-1,x+3,1) 10. The set of whole numbers 11. The set of natural numbers 12. The set of integers 13. Polynomials without a constant term 14. Thesetof rational numbers 15. The set of real numbers 16. Write About It Compare closure properties under the four operations for the set of rational numbers and the set of Irrational numbers.

1. (-1,0,1): Closed under addition and subtraction; not closed under multiplication and division. ### 2. \{0,8\}: Not closed under addition, subtraction, multiplication, or division. ### 3. \{x^2,1\}: Depends on x; generally not closed under any operation unless x=1. ### 4. \{0,x\}: Depends on x; generally not closed under any operation unless x=0. ### 5. \{-x^3,1,x^3\}: Depends on x; generally not closed under any operation unless x=0. ### 6. (-x,1,x+1): Depends on x; generally not closed under any operation unless x=0. ### 7. (-1,1): Closed under addition and subtraction; not closed under multiplication and division. ### 8. (-1,0,x]: Depends on x; generally not closed under any operation unless x=0. ### 9. (-1,x+3,1): Depends on x; generally not closed under any operation unless x=-2. ### 10. Whole numbers: Closed under addition and multiplication; not closed under subtraction and division. ### 11. Natural numbers: Closed under addition and multiplication; not closed under subtraction and division. ### 12. Integers: Closed under addition, subtraction, and multiplication; not closed under division. ### 13. Polynomials without a constant term: Closed under addition, subtraction, and multiplication; not closed under division. ### 14. Rational numbers: Closed under addition, subtraction, multiplication, and division (except by zero). ### 15. Real numbers: Closed under addition, subtraction, multiplication, and division (except by zero). ### 16. Rational vs Irrational Numbers: Rational numbers are closed under all operations except division by zero; irrational numbers are not closed under any operation. Explanation 1. Check Closure Under Addition A set is closed under addition if adding any two elements results in an element within the set. 2. Check Closure Under Subtraction A set is closed under subtraction if subtracting any two elements results in an element within the set. 3. Check Closure Under Multiplication A set is closed under multiplication if multiplying any two elements results in an element within the set. 4. Check Closure Under Division A set is closed under division if dividing any two elements (except by zero) results in an element within the set.