If a set of closed under addition, must it also be closed under multiplication? explain your answer or give a counterexample
if you mean the normal addition and multiplication used on whole numbers, then yes, since multiplication is just a shortcut for addition.
To determine whether a set that is closed under addition must also be closed under multiplication, we need to understand the concept of closure and examine both cases.
A set is said to be closed under an operation if applying that operation to any two elements in the set results in an element that also belongs to the same set.
Counterexample: Consider the set of negative integers, {-1, -2, -3, ...}. This set is closed under addition because adding any two negative integers always yields another negative integer. However, if we multiply two negative integers, such as -2 and -3, we get a positive result (6), which is not within the set. Hence, this set is not closed under multiplication.
Therefore, we can conclude that a set that is closed under addition does not necessarily have to be closed under multiplication, as demonstrated by the counterexample provided above.