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Mathematics and Statistics · Class 12 Commerce

Ch 1Mathematical Logic — Class 12 Mathematics and Statistics, concept-first.

Ordinary language is full of sentences whose truth cannot be settled — commands, questions, opinions and exclamations. Mathematical logic strips language down to just the sentences that can be judged true or false, and studies how such sentences combine.

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Key concepts

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Truth Tables of Compound Statements

A truth table lists every combination of truth values of the simple statements (2ⁿ rows for n statements) and the resulting value of the compound statement.

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In previous exams

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MHT-CET

1 concept · 4 appearances

2026, 2019

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

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Statements and Logical Connectives

Ordinary language is full of sentences whose truth cannot be settled — commands, questions, opinions and exclamations.

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Truth Tables of Compound Statements

The truth value of a compound statement is fixed entirely by the truth values of its simple parts and the connectives joining them.

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Statement Patterns: Tautology, Contradiction and Contingency

When simple statements are replaced by variables and combined by connectives, the resulting expression — for example — is called a statement pattern.

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Logical Equivalence

Two statement patterns are logically equivalent if they have identical final truth-table columns — that is, they take the same truth value for every assignment of truth values to their variables.

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Algebra of Statements and Duality

The logical equivalences of the previous section obey a tidy set of algebraic laws, closely parallel to the laws of ordinary algebra and of set theory.

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Negation of Compound Statements and Quantifiers

Negating a compound statement correctly is a skill in its own right, because the negation must move inside the connectives, not merely sit in front of the whole sentence.

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Application: Switching Circuits

The algebra of statements turns out to describe electrical switching circuits exactly, which is why logic underlies the design of every digital device.

Exercises

+Show 6 questions6 questions
  1. Q11Construct the truth table for $(p \vee q) \wedge \sim p$ and classify it as a tautology, contradiction or contingency.Free
  2. Q12Examine whether the statement pattern $\sim(p \wedge q) \vee p$ is a tautology.Free
  3. Q13The negation of the statement $p \rightarrow q$ is: (a) $\sim p \rightarrow \sim q$ (b) $p \wedge \sim q$ (c) $\sim p \vee q$ (d) $q \righta…Preview
  4. Q14The dual of the statement pattern $p \vee (q \wedge \sim r)$ is: (a) $p \wedge (q \vee \sim r)$ (b) $p \vee (q \wedge \sim r)$ (c) $\sim p \…Preview
  5. Q15Two switches connected in series in a circuit are represented symbolically by: (a) $p \vee q$ (b) $p \wedge q$ (c) $\sim p$ (d) $p \rightarr…Preview
  6. Q16Simplify the symbolic form of the switching circuit $(p \wedge q) \vee (\sim p \wedge q)$ using the algebra of statements, and state the sim…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 24 questions24 questions
  1. Q1If $p \vee q$ is true, then the truth value of $\sim p \wedge \sim q$ is ______.Preview
  2. Q2Write the converse, inverse, and contrapositive of the statement. "If $2 + 5 = 10$, then $4 + 10 = 20$."Preview
  3. Q3Determine whether the following statement pattern is a tautology, contradiction, or contingency: $[(\sim p \wedge q) \wedge (q \wedge r)] \w…Preview
  4. Q4The dual of the statement $(p \vee q) \wedge (r \vee s)$ is ______. (a) $(p \wedge q) \wedge (r \wedge s)$ (b) $(p \wedge q) \vee (r \wedge…Preview
  5. Q5Converse of the statement $q \rightarrow p$ is ______.Preview
  6. Q6Construct the truth table for the following statement pattern. $(p \wedge \sim q) \leftrightarrow (q \rightarrow p)$Preview
  7. Q7Write the negation of the following statement. $\exists n \in N, (n^2 + 2)$ is odd number.Preview
  8. Q8Write the negation of the following statement. Some continuous functions are differentiable.Preview
  9. Q9Write the negation of the following statement: $(p \rightarrow q) \vee (p \rightarrow r)$Preview
  10. Q10Which of the following is not a statement? (a) Smoking is injuries to health (b) $2 + 2 = 4$ (c) 2 is the only even prime number. (d) Come h…Preview
  11. Q11Examine whether the following statement pattern is a tautology, a contradiction or a contingency. $\sim p \rightarrow (p \rightarrow \sim q)…Preview
  12. Q12Consider the following statements. If D is dog, then D is very good. If D is very good, then D is dog. If D is not very good, then D is not…Preview
  13. Q13If $p$ : He is intelligent $q$ : He is strong Then, symbolic form of statement "It is wrong that, he is intelligent or strong" is: (a) $\neg…Preview
  14. Q14Write the converse, inverse, and contrapositive of the statement "If a triangle is equilateral, then it is equiangular."Preview
  15. Q15Using the truth table, verify. $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$Preview
  16. Q16U: the set of all real numbers. Q: the set of all rational numbers. I: the set of all integers. The above Venn diagram represents the truth…Preview
  17. Q17Statement $p \leftrightarrow q$ is true, when $p$ and $q$ have ______ truth values.Preview
  18. Q18Examine whether the following statement pattern is a tautology, a contradiction or a contingency. $(p \wedge \sim q) \rightarrow (\sim p \we…Preview
  19. Q19If $p$ : He swims $q$ : Water is warm Give the verbal statement for the following symbolic statement: $p \leftrightarrow \sim q$Preview
  20. Q20If $p$ : He swims $q$ : Water is warm Give the verbal statement for the following symbolic statement. $\sim (p \vee q)$Preview
  21. Q21If $p$ : He swims $q$ : Water is warm Give the verbal statement for the following symbolic statement. $q \rightarrow p$Preview
  22. Q22Write the negation of the following statements: (a) If it snows, then Gajashri does not drive car. (b) $\exists\, x \in \mathbb{N}$, such th…Preview
  23. Q23Let $p$ : Tanmay is a student $q$ : Tanmay likes to watch cricket match Write the verbal statement to describe each of the following: (a) $p…Preview
  24. Q24Using the truth table, verify $\sim(\sim p \rightarrow \sim q) \equiv \sim p \wedge q$.Preview

More questions

+Show 10 questions10 questions
  1. Example 1State which of the following are statements. For those that are, give the truth value. (i) $9$ is a prime number. (ii) Please bring me a gla…Free
  2. Example 2Let $p:$ "Riya studies hard" and $q:$ "Riya passes the examination". Write the following in symbolic form. (i) Riya studies hard and passes…Free
  3. Example 3Construct the truth table for the statement pattern $(p \wedge q) \rightarrow \sim p$ and hence state whether it is a tautology, a contradic…Free
  4. Example 4Show, using a truth table, that $(p \wedge q) \rightarrow p$ is a tautology.Preview
  5. Example 5Using truth tables, prove the logical equivalence $p \rightarrow q \equiv \sim p \vee q$.Preview
  6. Example 6Prove De Morgan's law $\sim(p \vee q) \equiv \sim p \wedge \sim q$ using truth tables.Preview
  7. Example 7Write the negation of the statement: "If it rains, then the match is cancelled."Preview
  8. Example 8Write the dual of the statement pattern $(p \wedge \sim q) \vee (\sim p \wedge q)$.Preview
  9. Example 9Write the negation of the quantified statement "$\forall x \in \mathbb{N},\ x + 3 > 2$", and state the truth value of the original statement…Preview
  10. Example 10Express the following switching circuit in symbolic form and simplify it: a branch with switches $p$ and $q$ in series, connected in paralle…Preview