📘 discrete mathematics

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

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. **Recurrence relation formula:**

1. **State the problem:** We have two relations $R_1$ and $R_2$ from set $A = B = \{1,2,3,4\}$ to itself. 2. **Define the relations:**

1. **Problem Statement:** Given sets $A = B = \{1,2,3,4\}$ and relations:

1. **Problem Statement:** Given sets $A = B = \{1,2,3,4\}$ and relations:

1. **Define with example:** (i) Finite and Infinite graphs:

1. The user provided an overview of a Discrete Mathematics course covering Boolean algebra, logic, set theory, combinatorics, graph theory, functions, and integers. 2. However, no

1. **State the problem:** We need to construct the intersection graph of the collection of sets $A_1, A_2, A_3, A_4, A_5$ where each vertex represents a set and an edge exists betw

1. Problem 7: Show that in any set of six classes, each meeting once a week on a weekday (Monday to Friday), there must be two classes meeting on the same day. 2. This is a classic

1. **Problem Statement:** Given the set $A = \{1,2,3,4\}$ and relation $R = \{(1,2), (2,3), (3,4), (1,4), (2,2), (3,3)\}$, determine if $R$ is reflexive, symmetric, antisymmetric,

1. முதலில், (A \times B) = 6 மற்றும் A = \{1, 3\} என்றால், \n(B) என்ன என்பதை காண்போம். 2. கார்டீசியன் தயாரிப்பின் அளவு \n(A \times B) = \n(A) \times \n(B) ஆகும்.

1. The problem is to define graphs using concepts from discrete mathematics, specifically elements of graph theory. 2. A graph $G$ is defined as an ordered pair $G = (V, E)$ where: