Define Each Of The Following Terms: (a) [3] DeMorgan’s Laws

Define each of the following terms: (a)[3] DeMorgan’s Laws for two primitive statements p and q say

DeMorgan’s Laws establish the relationships between negations of conjunctions and disjunctions of logical statements. Specifically, for two primitive statements p and q, these laws state that the negation of a conjunction is equivalent to the disjunction of the negations, and vice versa. Mathematically, they can be expressed as:

• ¬(p ∧ q) ≡ (¬p) ∨ (¬q)
• ¬(p ∨ q) ≡ (¬p) ∧ (¬q)

These principles are fundamental in propositional logic and are used extensively in logical reasoning, digital circuit design, and simplifying logical expressions.

Define each of the following terms: (b)[3] For two sets A and B, the Cartesian product A×B is...

The Cartesian product of two sets A and B, denoted as A×B, is the set of all ordered pairs where the first element belongs to A and the second element belongs to B. Formally:

• A×B = { (a, b) | a ∈ A, b ∈ B }

This operation forms the foundation for many concepts in set theory, relations, and functions, enabling the construction of ordered pairs essential in various mathematical and computational contexts.

Define each of the following terms: (c)[3] For a, b ∈ Z, m ∈ Z+, “a ≡ b mod(m)” means...

In modular arithmetic, the notation “a ≡ b mod(m)” indicates that integers a and b leave the same remainder when divided by m, or equivalently, that their difference is divisible by m. Formally:

• a ≡ b mod(m) ⇔ m | (a - b)

This relation partitions the set of integers into residue classes modulo m and is fundamental in number theory, cryptography, and various algorithms.

Define each of the following terms: (d)[3] A binary operation on a set A is...

A binary operation on a set A is a function that combines any two elements of A to produce another element of A. Formally, a binary operation * on A is a mapping:

• * : A × A → A

such that for any a, b ∈ A, the element a * b also belongs to A. Binary operations include addition, multiplication, and logical conjunction, and they form the basis for algebraic structures like groups, rings, and fields.

Define each of the following terms: (e)[3] A relation on A is anti-symmetric if...

A relation R on a set A is anti-symmetric if for all a, b ∈ A, whenever both (a, b) and (b, a) are in R, then a must be equal to b. Formally:

• ∀a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b.

Anti-symmetry is a key property in the definitions of partial orders and helps distinguish such relations from symmetric relations.

Define each of the following terms: (f)[3] A graph is planar if...

A graph is planar if it can be drawn on a two-dimensional plane without any edges crossing. Equivalently, a graph is planar if it can be embedded in the plane such that no two edges intersect except at their endpoints. Planarity is an important concept in graph theory with applications in circuit design, geography, and network visualization.

Define each of the following terms: (g)[3] A partition of a set X is...

A partition of a set X is a collection of non-empty, pairwise disjoint subsets of X whose union is X. In other words,:

• ⋃_i∈I A_i = X, and for all i ≠ j, A_i ∩ A_j = ∅, with each A_i ≠ ∅.

Partitions are used to divide sets into distinct classes for classification, equivalence relations, and other structured analyses.

Paper For Above instruction

In the field of mathematics and logic, foundational definitions such as those of DeMorgan’s Laws, Cartesian products, and modular congruences underpin the structure of formal reasoning and set theory. DeMorgan’s Laws, for example, are pivotal in simplifying logical expressions by relating negations of conjunctions and disjunctions, aiding both theoretical proofs and digital circuit design (Kleene, 1967). The Cartesian product provides a formal way to construct ordered pairs from two sets, forming the groundwork for defining relations and functions, which are essential in virtually all areas of mathematics and computer science (Epp, 2011). Within number theory, the concept of congruence modulo m captures the idea of equivalence classes of integers sharing the same remainder upon division by m, serving as a core operation in cryptography and computational algorithms (Sutherland, 2005). Binary operations extend the idea of combining elements within a set and are fundamental to algebraic structures such as groups and rings (Herstein, 2005). In graph theory, properties like planarity are central when visualizing networks, circuits, and geographic maps, as these properties determine how graphs can be embedded in the plane without crossings (West, 2001). Relations further define structural connections between elements of sets, with properties like anti-symmetry underpinning the concept of partial orders, which model hierarchy and precedence relations (Grätzer, 2003). Lastly, partitions describe how sets can be divided into disjoint subsets, facilitating the categorization of elements into classes that share common properties or equivalences, which are instrumental in the analysis of equivalence relations and quotient structures (Hungerford, 1974).

References

• Kleene, S. C. (1967). Mathematical Logic. Wiley.
• Epp, S. S. (2011). Discrete Mathematics with Applications. Brooks Cole.
• Sutherland, M. (2005). Introduction to Number Theory. Springer.
• Herstein, I. N. (2005). Topics in Algebra. Wiley.
• West, D. B. (2001). Introduction to Graph Theory. Prentice Hall.
• Grätzer, G. (2003). General Lattice Theory. Birkhäuser.
• Hungerford, T. W. (1974). Algebra. Springer.