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

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