1. **Problem a:** What is multiplicity of an element in a multiset? Find the multiplicities of each element in the multiset $\{a, a, a, \{a, a, a\}\}$. - The multiplicity of an element in a multiset is the number of times it appears. - Here, the multiset contains three $a$s and one element which is itself a multiset $\{a, a, a\}$. This inner multiset is considered as a single element different from $a$. - So, multiplicity of $a$ is 3, and multiplicity of $\{a, a, a\}$ is 1. 2. **Problem b:** Find $f^{-1} \circ g^{-1}$ where $f(x) = 2x + 1$ and $g(x) = \frac{x}{3}$. - Find $f^{-1}(y)$: Solve $y = 2x + 1 \Rightarrow x = \frac{y-1}{2}$. So, $f^{-1}(y) = \frac{y-1}{2}$. - Find $g^{-1}(z)$: Solve $z = \frac{x}{3} \Rightarrow x = 3z$. So, $g^{-1}(z) = 3z$. - Then, $f^{-1} \circ g^{-1} (x) = f^{-1}(g^{-1}(x)) = f^{-1}(3x) = \frac{3x - 1}{2}$. 3. **Problem c:** Let $P(S)$ be the power set of $S = \{a,b,c\}$. Determine the greatest lower bound (glb) and least upper bound (lub) of the set $\{\{a\}, \{c\}\}$ in poset $(P(S), \subseteq)$. - The glb (meet) is the intersection: $\{a\} \cap \{c\} = \emptyset$. - The lub (join) is the union: $\{a\} \cup \{c\} = \{a,c\}$. 4. **Problem d:** State the Modus Ponens rule of inference. - If $P \Rightarrow Q$ is true and $P$ is true, then $Q$ is true. 5. **Problem e:** How many generators are there of the cyclic group $G$ of order 6? - Number of generators equals $\varphi(6)$ where $\varphi$ is Euler's totient function. - $\varphi(6) = \varphi(2 \times 3) = \varphi(2) \times \varphi(3) = 1 \times 2 = 2$. - So, there are 2 generators. 6. **Problem f:** Define Normal subgroup. - A subgroup $N$ of group $G$ is normal if for every $g \in G$, $gNg^{-1} = N$. 7. **Problem g:** Obtain the generating function of the numeric function $a_r = 2^r + 3^r$, $r \geq 0$. - The generating function is $A(x) = \sum_{r=0}^\infty a_r x^r = \sum_{r=0}^\infty (2^r + 3^r) x^r = \sum_{r=0}^\infty (2x)^r + \sum_{r=0}^\infty (3x)^r$. - Using geometric series formula $\sum_{r=0}^\infty t^r = \frac{1}{1-t}$ for $|t|<1$, - So, $A(x) = \frac{1}{1-2x} + \frac{1}{1-3x}$.