For each of the following statements, determine whether the statement is true or false. for a true statement, give a proof. for a false statement, write out its negation and prove tha
a)for all rational numbers x, there is a positive integer n so that nx is an integer.
b) there is a positive integer n so that for all rational numbers x, nx is an integer

a) True

b) False

Step-by-step explanation:

a) True:

Let's set x as a rational number. We can rewrite x as p/q, with p an integer number and q a natural.

Let's notice that q is a positive integer (that's the definition of natural), so q*x = q*p/q = p is an integer.

q is the n we were looking for.

b) False:

Let's prove it by contradiction. We will assume there is a positive integer n that for all rational numbers x, n*x is an integer.

Let's take x = 1/n+1. By hypothesis, n*x = n*1/n+1 = n/n+1 has to be an integer.

The only way for that to be true is if n+1 = 1, but that means n = 0, that's absurd because n was a positive number.

That's why the statement is false.

m∠cdf = 43° , m∠gdf = 114°, m∠gde = 23°, m∠edh = 23° , m∠cdh = 157°

step-by-step explanation:

since, it is given that de bisects the angle gdh ⇒ de divides the angle gdh into two equal angles.

⇒ m∠gde = m∠edh

⇒ 8·x   - 1 = 6·x + 5

⇒ x = 3

so, m∠gde = 8·x - 1 = 23°

and m∠edh = 23°

now, it is given ∠cde is an straight angle so, m∠cde = 180°

⇒ m∠cdf + m∠fdg + m∠gde = 180°

⇒ 43° + m∠fdg + 23° = 180°

⇒ m∠fdg = 114°

again, m∠cde = m∠cdh + m∠edh = 180°

m∠cdh + 23° = 180°

⇒ m∠cdh = 157°

they go with it happens or they mite give a chance?

step-by-step explanation: