(ℤ, +, ·) has zero divisors?

In ℤ, the product of two non-zero integers is non-zero — so ℤ has no zero divisors. It's an integral domain.

If the data set {x_1, …, x_n} has standard deviation σ, the standard deviation of {2x_1, …, 2x_n} is:

σ(c · x) = |c| σ(x). With c = 2: σ(2x) = 2σ.

Problems & worked solutions

RO M1

A monoid is a semigroup that also has:

Show answer & worked solution

A monoid = semigroup + identity element. (A group adds the inverse-element axiom.)

RO M1

$(Z, +, \cdot)$ has zero divisors?

Show answer & worked solution

In $Z$ , the product of two non-zero integers is non-zero — so $Z$ has no zero divisors. It's an integral domain.

RO M1

If the data set ${x_{1}, \dots, x_{n}}$ has standard deviation $σ$ , the standard deviation of ${2 x_{1}, \dots, 2 x_{n}}$ is:

Show answer & worked solution

$σ (c \cdot x) = ∣ c ∣ σ (x)$ . With $c = 2$ : $σ (2 x) = 2 σ$ .

Problems & worked solutions

RO M1

A monoid is a semigroup that also has:

Show answer & worked solution

A monoid = semigroup + identity element. (A group adds the inverse-element axiom.)

RO M1

$(Z, +, \cdot)$ has zero divisors?

Show answer & worked solution

In $Z$ , the product of two non-zero integers is non-zero — so $Z$ has no zero divisors. It's an integral domain.

RO M1

If the data set ${x_{1}, \dots, x_{n}}$ has standard deviation $σ$ , the standard deviation of ${2 x_{1}, \dots, 2 x_{n}}$ is:

Show answer & worked solution

$σ (c \cdot x) = ∣ c ∣ σ (x)$ . With $c = 2$ : $σ (2 x) = 2 σ$ .