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SAT II Math II : Properties and Identities

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

Example Question #64 : Mathematical Relationships

What property of arithmetic is demonstrated below?

$\displaystyle y + 8 = 8 + y$

Possible Answers:

Identity

Distributive

Transitive

Associative

Commutative

Correct answer:

Commutative

Explanation:

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

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Example Question #65 : Mathematical Relationships

What property of arithmetic is demonstrated below?

$\displaystyle 0 + Z = Z$

Possible Answers:

Distributive

Reflexive

Identity

Inverse

Transitive

Correct answer:

Identity

Explanation:

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

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Example Question #66 : Mathematical Relationships

What property of arithmetic is demonstrated below?

$\displaystyle M + (-M) = 0$

Possible Answers:

Transitive

Symmetric

Associative

Inverse

Identity

Correct answer:

Inverse

Explanation:

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.

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Example Question #67 : Mathematical Relationships

What property of arithmetic is demonstrated below?

$m \angle ABC = m \angle ABC$ $\displaystyle m \angle ABC = m \angle ABC$

Possible Answers:

Symmetric

Commutative

Reflexive

Symmetric

Transitive

Correct answer:

Reflexive

Explanation:

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

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Example Question #68 : Mathematical Relationships

What property of arithmetic is demonstrated below?

If $m \angle 1 = m \angle 2$ $\displaystyle m \angle 1 = m \angle 2$ then $m \angle 2 = m \angle 1$ $\displaystyle m \angle 2 = m \angle 1$.

Possible Answers:

Commutative

Reflexive

Transitive

Associative

Symmetric

Correct answer:

Symmetric

Explanation:

If an equality is true, then it can be correctly stated with the expressions in either order with equal validity. This is the symmetric property of equality.

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Example Question #69 : Mathematical Relationships

What property of arithmetic is demonstrated below?

$m \angle ABC + 45^{\circ } = 45^{\circ } + m \angle ABC$ $\displaystyle m \angle ABC + 45^{\circ } = 45^{\circ } + m \angle ABC$

Possible Answers:

Transitive

Identity

Associative

Commutative

Distributive

Correct answer:

Commutative

Explanation:

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

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Example Question #7 : Properties And Identities

Which of the following sets is not closed under multiplication?

Possible Answers:

$\left \{ \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5} ... \right \}$ $\displaystyle \left \{ \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5} ... \right \}$

$\left \{ 1, 4, 9, 16, 25,...\right \}$ $\displaystyle \left \{ 1, 4, 9, 16, 25,...\right \}$

All of the sets in the other four responses are closed under multiplication.

$\left \{ i, 2i, 3i, 4i, 5i...\right \}$ $\displaystyle \left \{ i, 2i, 3i, 4i, 5i...\right \}$

$\left \{ 2, 4, 6, 8, 10...\right \}$ $\displaystyle \left \{ 2, 4, 6, 8, 10...\right \}$

Correct answer:

$\left \{ i, 2i, 3i, 4i, 5i...\right \}$ $\displaystyle \left \{ i, 2i, 3i, 4i, 5i...\right \}$

Explanation:

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.

$\left \{ 2, 4, 6, 8, 10...\right \}$ $\displaystyle \left \{ 2, 4, 6, 8, 10...\right \}$ is closed under multiplication since the product of two even numbers is even:

$2N \cdot 2M = 4MN = 2 (2MN)$ $\displaystyle 2N \cdot 2M = 4MN = 2 (2MN)$

$\left \{ 1, 4, 9, 16, 25,...\right \}$ $\displaystyle \left \{ 1, 4, 9, 16, 25,...\right \}$ is closed under multiplication since the product of two perfect squares is a perfect square:

$N^{2} \cdot M^{2} = (NM)^{2}$ $\displaystyle N^{2} \cdot M^{2} = (NM)^{2}$

$\left \{ \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5} ... \right \}$ $\displaystyle \left \{ \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5} ... \right \}$ is closed under multiplication since the product of two square roots of positive integers is the square root of a positve integer:

$\sqrt{N} \cdot \sqrt{M} = \sqrt{NM}$ $\displaystyle \sqrt{N} \cdot \sqrt{M} = \sqrt{NM}$

But $\left \{ i, 2i, 3i, 4i, 5i...\right \}$ $\displaystyle \left \{ i, 2i, 3i, 4i, 5i...\right \}$, as can be seen here, is not closed under multiplication:

$2i \cdot 3i = 2\cdot 3 \cdot i \cdot i = 6 \cdot i^{2} = 6 (-1)= -6 \notin \left \{ i, 2i, 3i, 4i, 5i...\right \}$ $\displaystyle 2i \cdot 3i = 2\cdot 3 \cdot i \cdot i = 6 \cdot i^{2} = 6 (-1)= -6 \notin \left \{ i, 2i, 3i, 4i, 5i...\right \}$

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Example Question #7 : Properties And Identities

Which expression is not equal to 0 for all positive values of $\displaystyle x$?

Possible Answers:

All four expressions given in the other choices are equal to 0 for all positive values of $\displaystyle x$.

-x + x $\displaystyle -x + x$

$\left (x+1 \right )^{0}$ $\displaystyle \left (x+1 \right )^{0}$

0 (x+1) $\displaystyle 0 (x+1)$

$0^{x}$ $\displaystyle 0^{x}$

Correct answer:

$\left (x+1 \right )^{0}$ $\displaystyle \left (x+1 \right )^{0}$

Explanation:

$\left (x+1 \right )^{0}$ $\displaystyle \left (x+1 \right )^{0}$ is the correct choice.

$\displaystyle 0 (x+1) = 0$ for all values of $\displaystyle x$, since, by the zero property of multiplication, any number multiplied by 0 yields product 0.

$\displaystyle -x + x = 0$ for all values of $\displaystyle x$ - this is a direct statement of the inverse property of addition.

$0^{x} = 0$ $\displaystyle 0^{x} = 0$, since 0 raised to any positive power yields a result of 0.

$\left (x+1 \right )^{0} = 1$ $\displaystyle \left (x+1 \right )^{0} = 1$, since any nonxero number raised to the power of 0 yields a result of 1.

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SAT II Math II : Properties and Identities

Study concepts, example questions & explanations for SAT II Math II

All SAT II Math II Resources

Example Questions

Example Question #64 : Mathematical Relationships

Example Question #65 : Mathematical Relationships

Example Question #66 : Mathematical Relationships

Example Question #67 : Mathematical Relationships

Example Question #68 : Mathematical Relationships

Example Question #69 : Mathematical Relationships

Example Question #7 : Properties And Identities

Example Question #7 : Properties And Identities

All SAT II Math II Resources

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