That any number is equal to itself is the reflexive property of equality.

The statement shows that two numbers can be multiplied in either order to yield the same product. This is the commutative property of multiplication.

The idea that one multiplied by any number yields the latter number as the product is the identity property of multiplication.

The statement demonstrates that two expressions both equal to a third expression are equal to each other. This is the transitive property of equality.

A set is closed under multiplication if and only the product of any two (not necessarily distinct) elements of that set is itself an element of that set.

is closed under multiplication since the product of two even numbers is even:

is closed under multiplication since the product of two perfect squares is a perfect square:

is closed under multiplication since the product of two square roots of positive integers is the square root of a positve integer:

But , as can be seen here, is not closed under multiplication:

is the correct choice.

for all values of , since, by the zero property of multiplication, any number multiplied by 0 yields product 0.

for all values of - this is a direct statement of the inverse property of addition.

, since 0 raised to any positive power yields a result of 0.

, since any nonxero number raised to the power of 0 yields a result of 1.

The statement shows that two numbers can be added in either order to achieve the same result. This is the commutative property of addition.

For every real number, there is a number that can be added to it to yield the sum 0. This is the inverse property of addition.

The fact that 0 can be added to any number to yield the latter number as the sum is the identity property of addition.

The statement shows that two numbers can be added in either order to yield the same sum. This is the commutative property of addition.