Question

The set 0, 2, 5 is under addition, as is illustrated by the fact that adding 2 and 5 results in 7, which is not included in the set. The set of all integers is closed under addition, as is illustrated by the fact that adding -20 and 6 results in -14, which is in the set. The set of all polynomials is under addition, which is illustrated by the fact that adding 3x^2 - 2 and 7x^4 + 4x results in 7x^4 + 3x^2 + 4x - 2, which is in the set.

a. True
b. False

Answer 1

The initial set of numbers is not closed under addition, while both sets of integers and all polynomials are closed under addition. This is exemplified by specific operations within each set that either produce or don't produce an element outside of the set.

Step-by-step explanation:

The set 0, 2, 5 is indeed not closed under addition as you can see that adding elements (like 2 and 5) would result in a number (7 in this case) that is not part of the set. On the contrary, the set of all integers is closed under addition, because any two integers added together will result in another integer, as illustrated by adding -20 and 6 to get -14, which is still an integer. Similarly, the set of all polynomials is closed under addition, as adding polynomials together results in another polynomial. An example provided: 3x^2 - 2 added to 7x^4 + 4x gives 7x^4 + 3x^2 + 4x - 2, which is indeed a polynomial.