Suppose ℤ denotes the set of all integers, ℤ+ denotes the set of all positive integers, and ℤ− denotes the set of all negative integers. similarly ℝ denotes the set of all real numbers, ℝ+ denotes the set of all positive real numbers, and ℝ− denotes the set of all negative real numbers. suppose ℕ denotes the set of all natural numbers and ℚ denotes the set of all rational numbers. enter "t" for each true, and "f" for each false statements.(a) z / q = z(b) z / z- = z+

a) False

b) False          

Step-by-step explanation:

We are given the following information in the question:

denotes the set of all integers.

denotes the set of all positive integers.

denotes the set of all negative integers.

denotes the set of all rational numbers

a) False

We will give a counter example .

b) False

We will give a counter example .

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.