Question

(a) Use a Venn diagram to illustrate the relationships A ⊆ B and B ⊆ C.
(b) Use a Venn diagram to illustrate the relationships A ⊂ B and B ⊂ C.
(c) Use a Venn diagram to illustrate the relationships A ⊂ B and A ⊂ C.
(d) Suppose that A , B, and C are sets such that A ⊆ B and B ⊆ C. Show that A ⊆ C.
(e) Find two sets A and B such that A ∈ B and A ⊆ B.

Answer 1

D)

A$\subset$ B, x$\in$ A$\to$ x$\isin$B,

B$\subset$C, x$\isin$ B$\to$ x$\isin$ C.

So for a particular X,x$\isin$ A$\to$ x$\isin$ B AND x$\isin$ B$\to$ X$\in$ C

By the transitivity of implication

this means that x$\isin$ A $\to$ x$\isin$ C

So A is subset of C.

E)A={} or $\emptyset$ and B={$\emptyset$ ,{1}}

A$\subset$ B,we can easily see that {} is an empty set which is a subset of B.

A$\isin$ B,since we can easily see that $\emptyset$ is an element of B.