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Please give an explanation and also tell me what does closed under addition and multiplication mean.

Different definitions are given everywhere.

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Could you mention some of those definitions? I only know one, but it seems like it would not help you to just state that if it happens to coincide with one of the ones you have already seen. – Tobias Kildetoft Jan 30 '13 at 18:30
How are you defining addition and multiplication here? I could interpret the set as being the integers mod 3 and then it would be closed since 1+1=-1. However, this may not be what others would infer at first glance. – JB King Jan 30 '13 at 18:38
1+1=2 ${}{}{}{}{}{}{}$ – user58512 Jan 30 '13 at 18:57

It means that if $a$ and $b$ are elements of the set, possibly equal, the sum $a+b$ and the product $ab$ are in the set.

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A set $X$ is closed under addition if $x+y\in X$ for any $x,y\in X$. It is closed under multiplication if $x\times y\in X$ for any $x,y\in X$. Note that $x$ and $y$ may or may not be equal.

The set $\{-1,0,1\}$ is closed under multiplication but not addition (if we take usual addition and multiplication between real numbers). Simply verify the definitions by taking elements from the set two at a time, possibly the same.

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A set $A$ is closed under a binary operation $\star: A \times A \longrightarrow A$ if for all $a, b \in A$, $a \star b \in A$. In other words, performing the binary operation on any two elements of the set always gives you back something that is also in the set.

So in this case, closure under addition means $a + b \in \{-1, 0, 1\}$ for all $a, b \in \{-1, 0, 1\}$. This set isn't closed under addition; consider $1 + 1$.

Closure under multiplication in this case means $ab \in \{-1, 0, 1\}$ for all $a, b \in \{-1, 0, 1\}$. This set is closed under multiplication; simply write out all possible products of elements.

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Let's go through 2 examples here and I'll show you how that set could be considered closed or not closed depending on the definition of the operations:

Integer mod 3 example:

In this case, there is {1,0,-1} which is equivalent to {1,0,2} mod 3 as 2=-1 in this case. In this case, the set is closed as all the operations stay in the field. In this case, 1+1=2 which is still in the field. 2+2 =4= 1 as remember that the equivalence here is mod 3, so a=b if there exists an integer k such that a = 3k+b.

Natural numbers case:

In this case, it isn't closed as 1+1=2 which isn't in the set.

This is why defining the operations is rather important as while most people can infer what addition and multiplication mean, they can be redefined in some cases.

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