Question

Which operations is the following set closed under: {2, 4, 6, ....}?

Answer 1

Multiplication 
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Answer 2

Answer: The answer is addition and multiplication.

Step-by-step explanation:  Closure property states that - for any numbers 'a' and 'b' belonging to a set A, if a*b also belongs to set A, then set A is said to be closed under the operation *.

The given set is X={2, 4, 6, . . .}. Let us check for the mathematical operations one by one.

(i) Addition: The addition of two even numbers is again an even number, so the given set is closed under addition. For example - 2+4=6, 4+6=10, etc.

(ii) Subtraction: The subtraction of two even numbers is either 0, an even number or a negative even number. So, X is not closed under subtraction. For example- 2-2=0, 4-2=2, 4-8=-4, etc.

(iii) Multiplication: The multiplication of two even numbers is again an even number. So, the set X is closed under multiplication. For example- 2×4=8, 6×4=24, etc.

(iv) Division: The division of two even numbers may or may not be an even number. So, X is not closed under division. For ex- 2/6=1/3, 6/8=3/4, etc.

Thus, the given set is closed under addition and multiplication.