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

Answer:

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

$x \in Y$

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