Question

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.

Answer 1

a. True

b. False

c. True

d. False

Explanation:

a. Meaning

as we can see that x is an element of Y

So notation is

`x \in Y`

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

So notation is

`X \subseteq Y`

2. (a) We need to proof

When
`A \subseteq B and B \subseteq C \so A \subseteq C`

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

When A = ∅ then
`A\subseteq C` 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

`x \in A`

As
`A \subseteq B,` each of the element of A is also an element of B

`x \in B`

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

`x \in C`

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

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
`A \in B` which is true

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

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

(c) We need to proof

When
`A \in B\ and \ B \subseteq C` then
`A \in C`

Let us assume that A, B and C are the sets that
`A\in B\ and\ B \subseteq C`

As,
`A \in B, \which\ is\ an\ element\ of\ the\ set\ B`

`A\in B`

As,
`B \subseteq C`, each of the element of B is also an element of C

`A\in 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
`A \in B` which is true

As {{1},2} is an element of B,
`B \in C` is correct

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

SO the statement is false