📘 discrete mathematics

1. **Problem 1:** Show that the relation $R$ on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b)R(c,d) \iff \frac{a}{c} = \frac{b}{d}$ is an equivalence relation. 2. **Recall the d

1. **State the problem:** We need to show that the relation $R$ on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b) R (c,d)$ if and only if $$ad(b+c) = bc(a+d)$$ is an equivalence

1. **Problem Statement:** We have a relation $R$ on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b) R (c,d) \iff ad = bc$ for all $(a,b), (c,d) \in \mathbb{N} \times \mathbb{N}$.

1. **State the problem:** We have a relation $R$ on $\mathbb{N} \times \mathbb{N}$ defined by

1. **Problem Statement:** We have a relation $R$ on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b) R (c,d) \iff ad = bc$ for all $(a,b),(c,d) \in \mathbb{N} \times \mathbb{N}$. W

1. **Problem statement:** Show that the relation $R$ on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b) R (c,d) \iff \frac{a}{c} = \frac{b}{d}$ is an equivalence relation. 2. **Re

1. **Problem:** Check whether the relation $S = \{(a,b) : a \leq b^3\}$ on $\mathbb{R}$ is reflexive, symmetric, or transitive. 2. **Reflexivity:** A relation $R$ on a set is refle

1. The problem shows a mapping from Set A to Set B with elements and their images. 2. Set A = \{4, -2, 8\} and Set B = \{3, 6, 7, 9\}.

1. **Problem statement:** Given the set $A := \{2, 9, 12, 25, 42, 100\}$ and relations $R_1$ and $R_2$ defined on $A$, analyze membership and equivalence classes as described. 2. *

1. The problem asks to identify a relation that is reflexive, anti-symmetric, and transitive. 2. Definitions:

1. The problem asks: What is a relation that is reflexive, anti-symmetric, and transitive called? 2. Recall the definitions:

1. The problem involves understanding the parity (odd or even) of numbers as represented in a binary tree structure. 2. Each node branches into two: one labeled "Odd" and the other

1. **State the problem:** We have a set $A = \{1,2,3,4,5,6\}$ and a relation $R$ defined by $R = \{(x,y) : y = x + 1, x,y \in A\}$. We want to find all pairs $(x,y)$ in $R$. 2. **U

1. **State the problem:** We have a discrete-time system described by the difference equation $$y(k+2) + 5y(k+1) + 6y(k) = U(k)$$ where $$U(k) = 1$$ for $$k \geq 0$$. We want to fi

1. **Problem Statement:** Given the Boolean function $$f(x,y,z) = (xy'z') + (y'z) + x'$$, find the truth table. 2. **Formula and Rules:** To find the truth table, evaluate $$f$$ fo

1. The problem asks: The solution to a homogeneous recurrence relation can be expressed in terms of which of the following? 2. A homogeneous recurrence relation is an equation that

1. **QUESTION ONE** 1.a. Problem: Today is Sunday, and neither this year nor next is a leap year. Find the day of the week one year from today.

1. **Problem 1a:** Given today is Sunday and neither this year nor next is a leap year, find the day of the week one year from today. - Since a non-leap year has 365 days, and 365

1. **Problem:** Given that today is Sunday and neither this year nor next is a leap year, what day of the week will be one year from today? 2. **Formula and rules:** The day of the

1. **Problem Statement:** Seven committees must elect a chairperson to represent them at the end-of-year board meeting. Some people serve on more than one committee and cannot be e

1. Let's start by understanding what a relation is: it's a set of ordered pairs, like (a, b), where a is related to b in some way. 2. A **reflexive relation** means every element i