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How do you prove the integers of the form 3n plus 1 is close under multiplication?
Suppose x and y are any two elements of the set.Then x = 3m + 1 and y = 3n + 1 for some integers m and n.
Then x*y = (3m + 1)*(3n + 1) = 9mn + 3m + 3n + 1 = 3*(3mn + m + n) + 1.
Since m and n are integers, mn+ m + n is also an integer = p, say.
Then x*y = 3p + 1.
That is the set is closed under multiplication.
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Is the set of even integers closed under multiplication?
Yes
Why are odd integers closed under multiplication but not under addition?
The numbers are not closed under addition because whole numbers, even integers, and natural numbers are closed.
Is set of integers is a group under multiplication?
No. One of the group axioms is that each element must have an inverse element. This is not the case with integers. In other words, you can't solve an equation like: 5 times "n" = 1 in the set of integers.
Is the set of positive integers a field under addition and multiplication?
NO. Certainly not. Additive inverse and Multiplicative inverse doesn't exist for many elements.
Which set is closed under the operation multiplication?
Any set where the result of the multiplication of any two members of the set is also a member of the set. Well known examples are: the natural numbers (ℕ), the integers (ℤ), the rational numbers (ℚ), the real numbers (ℝ) and the complex numbers (ℂ) - all closed under multiplication.
