Propositions, Truth Tables and Logical Connectives: Implication, Contrapositive, Converse and Necessary and Sufficient Conditions for SSC

Logical connectives and truth tables are the formal tools behind the conditional-statement questions in the reasoning section. This guide on propositions, truth tables and logical connectives for SSC explains the five main connectives with truth tables, the equivalent forms of an if–then statement, the negation rules, and the meaning of “only if”, “unless” and “necessary and sufficient”, with solved examples, practice questions and FAQs.

Propositions

A proposition (statement) is a sentence that is either true or false but not both. “Delhi is the capital of India” is a proposition; “Close the door” and “What time is it?” are not.

Logical Connectives

Truth Tables

For n statements, the truth table has 2ⁿ rows.

Forms Related to If P then Q

Also, P → Q is equivalent to ¬P ∨ Q. The negation of “If P then Q” is “P and not Q” (P ∧ ¬Q).

De Morgan’s Laws

Example: the negation of “It is hot and humid” is “It is not hot or it is not humid”. The negation of “He plays cricket or hockey” is “He does not play cricket and he does not play hockey”.

Tautology and Contradiction

• A tautology is always true, for example P ∨ ¬P.
• A contradiction is always false, for example P ∧ ¬P.
• A contingency is sometimes true and sometimes false.

Interpreting Conditional Words

Necessary and Sufficient Conditions

• Sufficient condition: if it is present, the result is guaranteed. Being a mother is sufficient for being a woman.
• Necessary condition: without it, the result cannot occur, but its presence alone does not guarantee the result. Being 18 is necessary for voting, but not sufficient (one must also be registered).
• Necessary and sufficient: the condition and the result always go together: P ↔ Q.

Solved Examples

Example 1. Find the contrapositive of “If it rains, the match will be cancelled”.
If the match is not cancelled, it did not rain.

Example 2. Find the converse of the same statement.
If the match is cancelled, it rained. (not equivalent)

Example 3. Find the negation of “If he studies, he will pass”.
He studies and he does not pass.

Example 4. “He will pass only if he studies.” Write it in the if–then form.
If he passes, then he studied (P → Q means passes → studies).

Example 5. “You will fail unless you work hard.” Write it in the if–then form.
If you do not work hard, you will fail.

Example 6. When is P → Q false?
Only when P is true and Q is false.

Example 7. How many rows will the truth table of three statements have?
2³ = 8.

Quick Tips

• Remember that an implication is false only when the antecedent is true and the consequent is false.
• Remember that only the contrapositive is equivalent to the original statement.
• Use De Morgan’s laws for the negation of “and” and “or”.
• Convert unless to If not.

Quick Revision Points

• P → Q is equivalent to ¬Q → ¬P.
• The converse and the inverse are equivalent to each other.
• ¬(P ∧ Q) = ¬P ∨ ¬Q.
• P only if Q means P → Q.
• P unless Q means if not Q then P.

Practice Questions With Answers

1. When is P → Q false?
   Answer: When P is true and Q is false
2. What is the contrapositive of If A then B?
   Answer: If not B then not A
3. What is the negation of P ∧ Q?
   Answer: ¬P ∨ ¬Q
4. What is the negation of If P then Q?
   Answer: P and not Q
5. How many rows are in the truth table of 4 statements?
   Answer: 16
6. What is P ∨ ¬P called?
   Answer: A tautology
7. What is the converse of If it is sunny, I will go out?
   Answer: If I go out, it is sunny
8. Translate: He will win only if he practises.
   Answer: If he wins, then he practised
9. Translate: Unless you hurry, you will be late.
   Answer: If you do not hurry, you will be late
10. When is P ↔ Q true?
    Answer: When P and Q have the same truth value

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Logical Reasoning notes page, read the previous topic Logical Fallacies and Common Reasoning Errors and the next topic Critical Reasoning: Paradox, Flaw and Evaluate.

Frequently Asked Questions

What is a proposition?

A sentence that is either true or false.

What is a conjunction?

The statement P and Q, true only when both are true.

What is a disjunction?

The statement P or Q, true when at least one is true.

What is an implication?

If P then Q, false only when P is true and Q is false.

What is the contrapositive?

If not Q then not P; it is equivalent to If P then Q.

What is the converse?

If Q then P; it is not equivalent to the original.

What are De Morgan’s laws?

The negation of an and statement is the or of the negations, and vice versa.

What is a tautology?

A statement that is always true.

What is a necessary condition?

A condition without which the result cannot occur.

What is a sufficient condition?

A condition that guarantees the result.

Is this topic asked in SSC?

Yes. It is the base of the conditional statement questions.

What does unless mean?

Unless Q, P means if not Q then P.