📘 discrete mathematics

1. **State the problem:** We have two relations \(\varphi = \{(1,a),(2,b),(3,c),(4,d),(5,g)\}\) and \(\sigma = \{(a,2),(b,1),(b,2),(e,3),(g,4)\}\). We need to find the compositions

1. **Problem statement:** Compute the solution of the recurrence relation $$a_n = 4a_{n-1} - 3a_{n-2} + 2^n + n + 3$$ with initial conditions $$a_0 = 1$$ and $$a_1 = 4$$.

1. **Problem Statement:** Given the set $A = \{3,6,9,12\}$ and a binary relation $R$ on $A$ defined by $R = \{(a,b) \mid a,b \in A, (a+b) \text{ divisible by } 6\}$.

1. Problem i: Produce the truth table and Boolean equation for the scenario: "If the room temperature is above a certain threshold (T) and the fan is not turned on (F'), and someon

1. **Problem statement:** Find the number of bytes (8-bit sequences) and answer related questions about bytes starting or ending with specific bit patterns.

1. The problem is to understand the sequence defined by $Y[n] = x[4n+1]$. 2. This means that the sequence $Y$ at index $n$ takes the value of the sequence $x$ at index $4n+1$.

1. **Problem statement:** We have a one-dimensional array $A$ with elements $A[1], A[2], \ldots, A[n]$ where $n = 50$. 2. **Part a:** How many elements are in the array?

1. **Problem Statement:** Determine whether the set of ordered pairs \(\{(b, a), (c, d), (d, a), (c, d), (a, d)\}\) is a function from \(W = \{a, b, c, d\}\) into \(W\). 2. **Defin

1. **Problem Statement:** Construct the Hasse diagram for the set $S=\{1,3,9,15,45,135\}$ with the relation defined by divisibility ($x \mid y$ means $x$ divides $y$). Also, determ

1. **Problem statement:** Given the set $A = \{1,2,3,4\}$ and the relation $R = \{(a,b) \mid a < b\}$ on integers, find: a) The inverse relation $R^{-1}$.

1. **Problem:** Let $R$ be a relation on the set $A = \{1,2,3,4,5,6\}$ defined by $R = \{(a,b) : a + b \leq 9\}$. **i) List the elements of $R$.**

1. **Problem statement:** Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps using strong induction. 2. **Base cases:** Check

1. **Problem:** Apply the Pigeonhole Principle to show that in any group of 13 people at least two were born in the same month. 2. **Pigeonhole Principle Statement:** If $n$ items

1. **Problem Statement:** Show that the sequence $\{a_n\}$ is a solution of the recurrence relation

1. **Problem Statement:** Prove that the relation $R$ on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b)R(c,d)$ if and only if $a + d = c + b$ is an equivalence relation. Also, fi

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 ele

1. **Problem 1: Membership in $\mathbb{Z}$** We need to determine if each object is an integer ($\in \mathbb{Z}$).

1. **Problem Statement:** Given the relation B on the set $\{1, 2, 3, 4, 5\}$ defined by the pairs $\{(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4)

1. **Problem:** Determine if the given graph has a Hamiltonian path and draw it. 2. **Understanding Hamiltonian Path:** A Hamiltonian path visits each vertex exactly once.

1. **Problem:** Decide whether each set of ordered pairs is a function from $W = \{a, b, c, d\}$ into $W$. A function must assign exactly one output in $W$ for each input in $W$. 2

1. **Problem:** Show whether $a_n = 2^{n+1} - 1$, $n \geq 1$, is a solution to the recurrence relation $a_n = 3a_{n-1} - 2a_{n-2}$. 2. **Formula:** The recurrence relation is given