Given sets A, B, U and a Venn diagram, answer questions about subsets, proper subsets, and elements.

Step 1: Address Question 1.
Given sets:
$A = \{x : x \in N, x \le 7\}$
$B = \{y : y \in N, y \le 5\}$
$U = \{0 \text{ देखि } 9 \text{ सम्मका संख्याहरु}\}$ (Numbers from 0 to 9)

• Natural numbers (N) are $\{1, 2, 3, ...\}$.
• Whole numbers (W) are $\{0, 1, 2, 3, ...\}$.

i) Write sets A, B, and U in tabular form.

$$A = \{1, 2, 3, 4, 5, 6, 7\}$$ $$B = \{1, 2, 3, 4, 5\}$$ $$U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$$

ii) Is set A a proper subset of B? Why or why not?
No, set A is not a proper subset of set B.
Reason: For A to be a proper subset of B, all elements of A must be in B, and B must contain at least one element not in A. However, elements 6 and 7 are in A but not in B.

iii) Write A and B in Venn diagram form.
To represent A and B in a Venn diagram:

• Draw a rectangle for the universal set U.
• Draw two circles inside U. Since all elements of B are also in A, set B is a subset of set A ($B \subset A$). Therefore, the circle representing B should be drawn entirely inside the circle representing A.
• Elements in B (which is also $A \cap B$): $\{1, 2, 3, 4, 5\}$
• Elements in A but not in B: $\{6, 7\}$
• Elements in U but not in A (and thus not in B): $\{0, 8, 9\}$

iv) Write all proper subsets of A.
Set $A = \{1, 2, 3, 4, 5, 6, 7\}$ has 7 elements. The number of proper subsets is $2^{|A|} - 1 = 2^7 - 1 = 128 - 1 = 127$.
Examples of proper subsets of A include:

$$\emptyset$$ $$\{1\}$$ $$\{1, 2\}$$ $$\{1, 2, 3, 4, 5, 6\}$$

... (There are 127 proper subsets in total.)

v) Write all improper subsets of B.
An improper subset of a set is the set itself.
The only improper subset of B is B itself:

$$\boxed{\{1, 2, 3, 4, 5\}}$$

Step 2: Address Question 2 based on the provided Venn diagram.
From the Venn diagram:
Universal Set $U = \{r, p, q, s, t, u, v\}$
Set $A = \{r, p, q\}$
Set $B = \{p, q, s\}$

i) Write the elements of sets A, B, and U in tabular form.

$$$$