Day 04Part12-Concept of tautology contradiction and Contigency, Satisfiable and Unsatisfiable case.

Day 04 – Part 12: Concept of Tautology, Contradiction, and Contingency

In Discrete Mathematics and Logic, statements can be categorized based on their truth values in all possible scenarios. The key concepts are Tautology, Contradiction, and Contingency, along with Satisfiable and Unsatisfiable cases.

1. Tautology (सर्वदा सत्य / Always True)

A tautology is a logical statement that is always true, regardless of the truth values of its individual components.

Example:

• $p\vee \neg p$ (Law of Excluded Middle)
• “It will either rain or not rain.” (Always True)

Truth Table:

$p$	$\neg p$	$p\vee \neg p$
T	F	T
F	T	T

Since the last column contains all True (T) values, it is a Tautology.

2. Contradiction (सर्वदा असत्य / Always False)

A contradiction is a logical statement that is always false, regardless of the truth values of its components.

Example:

• $p\wedge \neg p$ (Self-Contradiction)
• “It is raining and not raining at the same time.” (Always False)

Truth Table:

$p$	$\neg p$	$p\wedge \neg p$
T	F	F
F	T	F

Since the last column contains all False (F) values, it is a Contradiction.

3. Contingency (कभी सत्य, कभी असत्य / Sometimes True, Sometimes False)

A contingency is a logical statement that is sometimes true and sometimes false, depending on the values of its components.

Example:

• $p\vee q$ (Logical OR between two statements)

Truth Table:

$p$	$q$	$p\vee q$
T	T	T
T	F	T
F	T	T
F	F	F

Since the last column has both True (T) and False (F) values, it is a Contingency.

4. Satisfiable and Unsatisfiable Cases

Satisfiable Statement:

• A proposition is satisfiable if it is true in at least one case.
• Example: $p\vee q$ (It has at least one True in the truth table).

Unsatisfiable Statement:

• A proposition is unsatisfiable if it is never true (always false, i.e., contradiction).
• Example: $p\wedge \neg p$ (Always False).

Summary

Concept	Definition	Example
Tautology	Always True	$p\vee \neg p$
Contradiction	Always False	$p\wedge \neg p$
Contingency	Sometimes True, Sometimes False	$p\vee q$
Satisfiable	True in at least one case	$p\vee q$
Unsatisfiable	Never True (Always False)	$p\wedge \neg p$

Would you like more examples or practical applications of these concepts?

Day 04Part12-Concept of tautology contradiction and Contigency , Satisfiable and Unsatisfiable case.

Discrete Mathematics and Its Applications, Eighth Edition

1.3. Tautology /Contradiction / Contingency

1.3 Propositional Equivalences

DISCRETE MATHEMATICS

Here is a clear and complete explanation of:

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Day 04 Part-12: Concepts of Tautology, Contradiction, Contingency, Satisfiable & Unsatisfiable Statements

(Discrete Mathematics / Logic – for GATE, BCA, B.Tech, CS)

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1. Propositional Logic Recap

A proposition is a statement that is either True (T) or False (F).

Examples:

• “It is raining.” (Proposition)
• “x + 3 = 7” (Not a proposition until x is known)

We use logical connectives like:

• ∧ (AND), ∨ (OR), → (IMPLIES), ¬ (NOT)

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2. Tautology (Always True)

A tautology is a compound statement that is always true for all possible truth values of its variables.

Example:

(P ∨ ¬P) is always true.

P	¬P	P ∨ ¬P
T	F	T
F	T	T

Output is always True → It’s a tautology

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3. Contradiction (Always False)

A contradiction is a compound statement that is always false, no matter what the input values are.

Example:

(P ∧ ¬P) is always false.

P	¬P	P ∧ ¬P
T	F	F
F	T	F

Output is always False → It’s a contradiction

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4. Contingency (Sometimes True, Sometimes False)

A contingency is a compound statement that is true for some cases and false for others.

Example:

(P ∧ Q)

P	Q	P ∧ Q
T	T	T
T	F	F
F	T	F
F	F	F

Sometimes True → It’s a contingency

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5. Satisfiable Statement

A formula is satisfiable if at least one combination of truth values makes the formula true.

Example: (P ∨ Q)

• This is satisfiable, because in cases like P=T, Q=F → Output is T

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6. Unsatisfiable Statement

A formula is unsatisfiable if no combination of values makes it true → same as contradiction

Example: (P ∧ ¬P)

• Always false → unsatisfiable

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Summary Table

Concept	Definition	Example
Tautology	Always true	P ∨ ¬P
Contradiction	Always false	P ∧ ¬P
Contingency	Sometimes true, sometimes false	P → Q
Satisfiable	At least one case is true	P ∨ Q
Unsatisfiable	No case is true	P ∧ ¬P

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Day 04Part12-Concept of tautology contradiction and Contigency , Satisfiable and Unsatisfiable case.