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

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step-by-step explanation:

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