1. **Discuss all features of a graph:** A graph consists of vertices (nodes) and edges (connections between nodes). Key features include: - **Degree of a vertex:** Number of edges incident to the vertex. - **Simple graph:** No loops or multiple edges between the same vertices. - **Connectedness:** Whether there is a path between every pair of vertices. - **Cycles:** Paths that start and end at the same vertex without repeating edges. - **Components:** Maximal connected subgraphs. - **Complete graph:** Every pair of distinct vertices is connected by an edge. 2. **Draw graphs with specified properties:** (i) Graph with four vertices of degrees 1,1,1, and 4: - Let vertices be A, B, C, D. - Vertex D has degree 4, so it connects to all other three vertices plus one more edge (loop not allowed in simple graph, so connect D to A, B, C and add an edge between two of A, B, C to increase D's degree to 4). - Degrees: A=1, B=1, C=1, D=4. (ii) Simple graph with nine edges and all vertices of degree 3: - Let number of vertices be $n$. - Sum of degrees = $3n$. - By Handshaking Lemma, sum of degrees = $2 imes$ number of edges. - So, $3n = 2 imes 9 = 18 ightarrow n=6$ vertices. - Construct a 6-vertex graph where each vertex has degree 3. (iii) Graph with five vertices of degrees 1, 2, 3, 3, and 5: - Sum of degrees = $1+2+3+3+5=14$. - Number of edges = $14/2=7$. - Construct a graph with 5 vertices and 7 edges matching these degrees. 3. **Prove that a necessary and sufficient condition for a non-negative integer $n$ to be divisible by a positive integer $d$ is that $n \bmod d = 0$:** - **Necessary:** If $n$ is divisible by $d$, then $n = kd$ for some integer $k$. - By division algorithm, $n = dq + r$ with $0 \leq r < d$. - Since $n = kd$, remainder $r=0$, so $n \bmod d = 0$. - **Sufficient:** If $n \bmod d = 0$, remainder $r=0$, so $n = dq$ for some integer $q$. - Hence, $n$ is divisible by $d$. 4. **Given integers $a,b,c$, if $a-b$ is even and $b-c$ is even, what can you say about parity of $2a - (b+c)$? Prove:** - Since $a-b$ and $b-c$ are even, $a \equiv b \pmod{2}$ and $b \equiv c \pmod{2}$. - So, $a \equiv c \pmod{2}$. - Consider $2a - (b+c)$ modulo 2: $$2a - (b+c) \equiv 0 - (b+c) \equiv -(b+c) \equiv b+c \pmod{2}$$ - Since $a \equiv b \equiv c \pmod{2}$, $b+c \equiv a+a = 2a \equiv 0 \pmod{2}$. - Therefore, $2a - (b+c)$ is even. 5. **Indicate whether the given subsets of real numbers are commutative rings:** (a) Set of all even numbers: - Closed under addition and multiplication. - Contains additive identity 0. - Additive inverses exist (negative even numbers). - Multiplication is commutative. - Hence, it forms a commutative ring. (b) Set of all nonnegative real numbers: - Closed under addition and multiplication. - Contains additive identity 0. - Additive inverses do not exist (no negatives). - Hence, not a ring. (c) Set of all rational numbers: - Closed under addition and multiplication. - Contains additive identity 0. - Additive inverses exist. - Multiplication is commutative. - Hence, it forms a commutative ring.