For each of the following statements about sets A, B, and C, either prove the statement is true or give a counterexample to show that it is false. a. If A⊆BA⊆B and B⊆CB⊆C, then A⊆CA⊆C.
b. If A∈BA∈B and B⊆CB⊆C, then A⊆CA⊆C.
c. If A∈BA∈B and B⊆CB⊆C, then A∈CA∈C.
d. If A∈BA∈B and B∈CB∈C, then A∈CA∈C.

a. True

b. False

c. True

d. False

Step-by-step explanation:

a. Meaning

as we can see that x is an element of Y

So notation is

Therefore x is the subset of Y if each of an element of X is also an element of Y.

So notation is

2. (a) We need to proof

When

We will say that A, B and C are the set that is

When A = ∅ then which shows true, as the set of empty is a subset of each set.

Hence, it is safe to say that A is not the empty set.

Now we will proof directly

Let us say x be an element of A

As each of the element of A is also an element of B

As , each if the element of B is also an element of C

Therefore, as we can see that each of an element of A is also known an element of C, that states

So, the given statement is true, as we conclude with a proof.

(b). We will assume {1}, B = {{1},2} and C = {{1},2,3}

As in the point a, which is an element of B, that is which is true

As all of the elements in B are also an element in C, which is also correct.

Although, is false as 1 is an element of A that is not in C.

(c) We need to proof

When   then

Let us assume that A, B and C are the sets that

As,

As, , each of the element of B is also an element of C

So, its true.

(d) We will assume {1}, B = {{1},2} and C = {{1},2,3}

As in the point a, which is an element of B, that is which is true

As {{1},2} is an element of B, is correct

Although is not correct as {1) is not an element in C.

SO the statement is false

x = -3

y = -18

step-by-step explanation:

to solve by substitution, substitute one variable and solve for the other. then substitute again to find the remaining variable.

x-1/3(3x-9) = -3 since y=3x-9 substittute y.

x-1/3(3x-9)=-3

x + x +3 = -3

2x+3=-3

2x+3-3= -3-3

2x=-6

x=-3

substitute again into y=3x-9

y=3(-3)-9

y=-9-9

y=-18

option a is correct.

step-by-step explanation:

scale factor: it is used to find proportional measurements.

proportions means having the same ratio.

scale factor is the ratio of the model measurement to the actual measurement in a simplest form.

or we can say that the ratio of any two corresponding lengths in two similar geometric figures

if the scale factor is more than 1, then it is an enlargement,

if the scale factor less than 1, then it is a reduction.

given the two geometric figure:

by definition of proportions, we have