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Which set is closed under the given operation 1 integers under division 2 negative integers under subtraction 3 odd integers under multiplication?
1 No. 2 No. 3 Yes.
Are odd numbered sets closed under multiplication?
Yes
Are odd numbered sets closed under addition and multiplication?
No. 1 + 3 = 4, which is not odd. In fact, no pair of odds sums to an odd. So the set is not closed under addition.
Is the set of odd numbers is closed under multiplication but is not under addition?
Let + (addition) be a binary operation on the set of odd numbers S. The set S is closed under + if for all a, b ϵ S, we also have a + b ϵ S. Let 3, 5 ϵ the set of odd numbers 3 + 5 = 8 (8 is not an odd number) Since 3 + 5 = 8 is not an element of the set of the odd numbers, the set of the odd numbers is not closed under addition.
Why is an odd number times an odd number an odd number?
Suppose m and n are integers. Then 2m + 1 and 2n +1 are odd integers.(2m + 1)*(2n + 1) = 4mn + 2m + 2n + 1 = 2*(2mn + m + n) + 1 Since m and n are integers, the closure of the set of integers under multiplication and addition implies that 2mn + m + n is an integer - say k. Then the product is 2k + 1 where k is an integer. That is, the product is an odd number.
