📘 discrete mathematics

1. The problem is to determine if the given relation \( f \) from set \( D = \{1, 2, 3, 4\} \) to set \( Y = \{a, b, c, d\} \) defined by: \( f(1) = a, f(2) = b, f(3) = b, f(4) = c

1. **Problem statement:** Given the relation $R$ on $\mathbb{N}$ defined by $xRy \iff \frac{2x + y}{3} \in \mathbb{N}$, check pairs, prove $R$ is an equivalence relation, and find

1. The problem asks us to draw the Hasse diagram for the partially ordered set $(\{1, 2, 3, 4, 6, 8, 12\}, \mid)$ where the order relation is divisibility: $a \leq b$ if and only i

1. **State the problem:** We are asked to draw the Hasse diagram for the partial order defined by the divisibility relation $a \mid b$ on the set $\{1, 2, 3, 4, 6, 8, 12\}$.

1. **Problem 1: Number Systems in Computing** a) The binary, octal, decimal, and hexadecimal number systems are fundamental in computing because they uniquely suit digital electron

1. Given sets and relations: \(R = \{(1,2),(1,3),(2,3),(3,1),(3,3)\}\)

1. Stating the problem: We have relations $R$ and $S$ on the set $\{1,2,3\}$: $$R=\{(1,2),(1,3),(2,3),(3,1),(3,3)\}$$

1. Consider the problem involving set operations and with conditions: Given:

1. The problem is to find the power set of a given set, but the set itself was not specified in your question. 2. The power set of any set $S$ is the set of all possible subsets of

1. The problem asks to state the Boolean expression laws. 2. Boolean algebra has several fundamental laws that govern the operations AND ($\cdot$), OR ($+$), and NOT ($\overline{\c

1. **Construct a truth table for the compound proposition $(P \to R) \land (Q \lor \neg R)$**. - List all possible truth values for $P, Q, R$.

1. **Problem statement:** Given sets and relations, solve parts (a) to (f). **(a)(i) Find the composition relation $R \circ S$ where $R: A \to B$, $S: B \to C$:

1. **Show that $t \wedge s$ can be derived from the premises $p \to q$, $q \to \neg r$, $r$, $p \lor (t \wedge s)$.** - From $r$ and $q \to \neg r$, since $r$ is true, $q$ must be

1. Problem (a): Find the power set of $A = \{a,b,c,d,e\}$. The power set is the set of all subsets of $A$.

1. **Problem:** Prove that a connected graph is Eulerian if and only if all its vertices have even degree. **Step 1:** Recall the definition: A graph is Eulerian if it contains a c

1. Discrete mathematics is a branch of mathematics dealing with discrete elements that uses algebra and arithmetic. 2. Common topics include logic, set theory, combinatorics, graph

1. The problem asks to identify properties of relation $$R = \{(1,2), (2,2), (1,1), (4,4), (1,3), (3,3), (3,2)\}$$ on set $$A=\{1,2,3,4\}$$. \nCheck reflexivity: Elements 1, 2, 3,

1. **Solve the recurrence relation** $a_n = 8a_{n-1} - 16a_{n-2}$ for $n\ge 2$ with $a_0=16, a_1=80$. 2. Write the characteristic equation: $$r^2 - 8r + 16 = 0$$

1. Problem (7a): Solve the recurrence relation $$a_n = 8a_{n-1} - 16a_{n-2}$$ for $$n \geq 2$$ with initial conditions $$a_0 = 16$$ and $$a_1 = 80$$. 2. Step 1: Characteristic equa

1. Problem: Solve the recurrence relation $a_n = 8a_{n-1} - 16a_{n-2}$ for $n \ge 2$ with initial conditions $a_0 = 16$ and $a_1 = 80$. Step 1: Write the characteristic equation as