MATH SOLVE

3 months ago

Q:
# PLEASE HELP ASAP Select from the drop-down menus to correctly identify whether the given operation is closed or not closed with respect to each set of numbers. Real numbers are (closed,not closed) under addition Integers are ( closed, not closed) under division.Irrational numbers are (closed, not closed) under multiplication.Rational numbers are ( closed, not closed) under subtraction.

Accepted Solution

A:

Real numbers are "closed" under addition
Integers are "not closed" under division.
Irrational numbers are "not closed" under multiplication.
Rational numbers are "closed" under subtraction.
If an operation is closed under a set of numbers that means that the result of the operation will always be within that same set of numbers. If that's not true, then the operation is not closed. So with that in mind, let's look at the problems.
Real numbers are (closed,not closed) under addition
* Since a real number plus a real number always results in another real number, then real numbers are closed under addition. So the answer is "closed"
Integers are ( closed, not closed) under division.
* Let's try this counter example. 1 divided by 2 = 1/2. 1/2 is NOT an integer, therefore integers are not closed under division. The answer is "not closed"
Irrational numbers are (closed, not closed) under multiplication.
* This is a tricky one. You may give an impulsive answer and say "closed". After all, how could you possibly multiply one non repeating infinite sequence by another and get something rational?. But what's pi multiplied by the reciprocal of pi? Both pi and 1/pi are irrational. Yet when you multiply them together you get 1 which is quite rational. So the answer is "not closed"
Rational numbers are ( closed, not closed) under subtraction.
* Since rational numbers are all numbers that can be expressed as a fraction consisting on an integer numerator and divisor, we can express subtraction as a/b - c/d = ad/bd - bc/bd = (ad-bc)/bd and since all we're doing is adding, subtracting, and multiplying integers which is closed under those operations, that means that rational numbers are also closed under subtraction. So the answer is "closed".