Skip to main content

Discrete Mathematics -Propositional Equivalences


Propositional Equivalences

Two statements X and Y are logically equivalent if any of the following two conditions hold −
  • The truth tables of each statement have the same truth values.
  • The bi-conditional statement X⇔Y is a tautology.
Example − Prove ¬(A∨B)and[(¬A)∧(¬B)] are equivalent

Testing by 1st method (Matching truth table)

ABA ∨ B¬ (A ∨ B)¬ A¬ B[(¬ A) ∧ (¬ B)]
TrueTrueTrueFalseFalseFalseFalse
TrueFalseTrueFalseFalseTrueFalse
FalseTrueTrueFalseTrueFalseFalse
FalseFalseFalseTrueTrueTrueTrue
Here, we can see the truth values of ¬(A∨B)and[(¬A)∧(¬B)] are same, hence the statements are equivalent.

Testing by 2nd method (Bi-conditionality)

AB¬ (A ∨ B )[(¬ A) ∧ (¬ B)][¬ (A ∨ B)] ⇔ [(¬ A ) ∧ (¬ B)]
TrueTrueFalseFalseTrue
TrueFalseFalseFalseTrue
FalseTrueFalseFalseTrue
FalseFalseTrueTrueTrue
As [¬(A∨B)]⇔[(¬A)∧(¬B)] is a tautology, the statements are equivalent.

Inverse, Converse, and Contra-positive

Implication / if-then (→) is also called a conditional statement. It has two parts −
  • Hypothesis, p
  • Conclusion, q
As mentioned earlier, it is denoted as p→q.
Example of Conditional Statement − “If you do your homework, you will not be punished.” Here, "you do your homework" is the hypothesis, p, and "you will not be punished" is the conclusion, q.
Inverse − An inverse of the conditional statement is the negation of both the hypothesis and the conclusion. If the statement is “If p, then q”, the inverse will be “If not p, then not q”. Thus the inverse of p→q is ¬p→¬q.
Example − The inverse of “If you do your homework, you will not be punished” is “If you do not do your homework, you will be punished.”
Converse − The converse of the conditional statement is computed by interchanging the hypothesis and the conclusion. If the statement is “If p, then q”, the converse will be “If q, then p”. The converse of p→q is q→p.
Example − The converse of "If you do your homework, you will not be punished" is "If you will not be punished, you do your homework”.
Contra-positive − The contra-positive of the conditional is computed by interchanging the hypothesis and the conclusion of the inverse statement. If the statement is “If p, then q”, the contra-positive will be “If not q, then not p”. The contra-positive of p→q is ¬q→¬p.
Example − The Contra-positive of " If you do your homework, you will not be punished” is "If you are punished, you did not do your homework”.

Duality Principle

Duality principle states that for any true statement, the dual statement obtained by interchanging unions into intersections (and vice versa) and interchanging Universal set into Null set (and vice versa) is also true. If dual of any statement is the statement itself, it is said self-dual statement.
Example − The dual of (A∩B)∪C is (A∪B)∩C

Normal Forms

We can convert any proposition in two normal forms −
  • Conjunctive normal form
  • Disjunctive normal form

Conjunctive Normal Form

A compound statement is in conjunctive normal form if it is obtained by operating AND among variables (negation of variables included) connected with ORs. In terms of set operations, it is a compound statement obtained by Intersection among variables connected with Unions.
Examples
  • (A∨B)∧(A∨C)∧(B∨C∨D)
  • (P∪Q)∩(Q∪R)

Disjunctive Normal Form

A compound statement is in conjunctive normal form if it is obtained by operating OR among variables (negation of variables included) connected with ANDs. In terms of set operations, it is a compound statement obtained by Union among variables connected with Intersections.
Examples
  • (A∧B)∨(A∧C)∨(B∧C∧D)

Comments

Post a Comment

Popular posts from this blog

Discrete Mathematics - Rules of Inference

To deduce new statements from the statements whose truth that we already know,  Rules of Inference  are used. What are Rules of Inference for? Mathematical logic is often used for logical proofs. Proofs are valid arguments that determine the truth values of mathematical statements. An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises (or hypothesis). The symbol “ ∴ ∴ ”, (read therefore) is placed before the conclusion. A valid argument is one where the conclusion follows from the truth values of the premises. Rules of Inference provide the templates or guidelines for constructing valid arguments from the statements that we already have. Table of Rules of Inference Rule of Inference Name Rule of Inference Name P ∴ P ∨ Q P ∴ P ∨ Q Addition P ∨ Q ¬ P ∴ Q P ∨ Q ¬ P ∴ Q Disjunctive Syllogism P Q ∴ P ∧ Q P Q ∴ P ∧ Q Conjunction P → Q Q → R ∴ P → R P → Q Q → R ∴ P → R ...
Post a Comment
Read more

discrete mathematics:Introduction to Trees

Tree  is a discrete structure that represents hierarchical relationships between individual elements or nodes. A tree in which a parent has no more than two children is called a binary tree. Tree and its Properties Definition  − A Tree is a connected acyclic undirected graph. There is a unique path between every pair of vertices in  G G . A tree with N number of vertices contains  ( N − 1 ) ( N − 1 )  number of edges. The vertex which is of 0 degree is called root of the tree. The vertex which is of 1 degree is called leaf node of the tree and the degree of an internal node is at least 2. Example  − The following is an example of a tree − Centers and Bi-Centers of a Tree The center of a tree is a vertex with minimal eccentricity. The eccentricity of a vertex  X X  in a tree  G G  is the maximum distance between the vertex  X X  and any other vertex of the tree. The maximum eccentricity is the tree diameter. If a ...
Post a Comment
Read more

Digital Circuits - Decoders

Decoder  is a combinational circuit that has ‘n’ input lines and maximum of 2 n output lines. One of these outputs will be active High based on the combination of inputs present, when the decoder is enabled. That means decoder detects a particular code. The outputs of the decoder are nothing but the  min terms  of ‘n’ input variables (lines), when it is enabled. 2 to 4 Decoder Let 2 to 4 Decoder has two inputs A 1  & A 0  and four outputs Y 3 , Y 2 , Y 1  & Y 0 . The  block diagram  of 2 to 4 decoder is shown in the following figure. One of these four outputs will be ‘1’ for each combination of inputs when enable, E is ‘1’. The  Truth table  of 2 to 4 decoder is shown below. Enable Inputs Outputs E A 1 A 0 Y 3 Y 2 Y 1 Y 0 0 x x 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 From Truth table, we can write the  Boolean functions  for each output as Y 3 = E . A 1 . A 0 Y 3 = E . A ...
Post a Comment
Read more