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Mathematics · Class 12 Science

Ch 12Discrete Mathematics — Class 12 Mathematics, concept-first.

Mathematics is often split into two broad styles. Continuous mathematics works with the set of real numbers, which is uncountably infinite -- between any two real numbers there is always another whole uncountable set of numbers, so a continuous function can be sketched as a smooth, unbroken curve.

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Binary Operations

A binary operation on a set assigns to each ordered pair an element (closure). Its algebraic properties are: - Commutative: for all . - Associative: for all . - Identity element : for all .

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Chapter contents

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

12.1

Introduction

Mathematics is often split into two broad styles. Continuous mathematics works with the set of real numbers, which is uncountably infinite -- between any two real numbers there is always another whole…

12.2

Binary Operations

The word "operation" refers to a process performed on either a single element or several elements at once. Finding the negative of a real number, , needs just one input -- it is an unary operation.

12.2.1

Definitions

Starting from a familiar example. Take the ordinary addition and multiplication of natural numbers, and for .

12.2.2

Some More Properties of a Binary Operation

Beyond closure, a binary operation on a set may or may not satisfy four further properties.

12.2.3

Some Binary Operations on Boolean Matrices

Definition 12.3. A Boolean matrix is a real matrix whose every entry is either or . Boolean entries naturally model "off/on" in electrical switching circuits, or the adjacency matrix of a graph.

12.2.4

Modular Arithmetic

Modular arithmetic replaces an integer by the remainder it leaves on division by a fixed modulus , written (" is congruent to modulo ").

+Exercise 12.1i10 questions
  1. Q1Determine whether $*$ is a binary operation on the sets given below. (i) $a*b = a\cdot|b|$ on $\mathbb{R}$ (ii) $a*b = \min(a,b)$ on $A=\{1,…Free
  2. Q2On $\mathbb{Z}$, define $*$ by $(m*n) = m^n + n^m,\ \forall\, m,n \in \mathbb{Z}$. Is $*$ binary on $\mathbb{Z}$?Free
  3. Q3Let $*$ be defined on $\mathbb{R}$ by $(a*b) = a+b+ab-7$. Is $*$ binary on $\mathbb{R}$? If so, find $3*\left(\dfrac{-7}{15}\right)$.Free
  4. Q4Let $A=\{a+\sqrt5\,b : a,b\in\mathbb{Z}\}$. Check whether the usual multiplication is a binary operation on $A$.Preview
  5. Q5(i) Define an operation $*$ on $\mathbb{Q}$ as follows: $a*b=\left(\dfrac{a+b}{2}\right);\ a,b\in\mathbb{Q}$. Examine the closure, commutati…Preview
  6. Q6Fill in the following table so that the binary operation $*$ on $A=\{a,b,c\}$ is commutative. | $*$ | $a$ | $b$ | $c$ | |---|---|---|---| |…Preview
  7. Q7Consider the binary operation $*$ defined on the set $A=\{a,b,c,d\}$ by the following table: | $*$ | $a$ | $b$ | $c$ | $d$ | |---|---|---|--…Preview
  8. Q8Let $A=\begin{pmatrix}1&0&1&0\\0&1&0&1\\1&0&0&1\end{pmatrix}$, $B=\begin{pmatrix}0&1&0&1\\1&0&1&0\\1&0&0&1\end{pmatrix}$, $C=\begin{pmatrix}…Preview
  9. Q9(i) Let $M=\left\{\begin{pmatrix}x&x\\x&x\end{pmatrix} : x\in\mathbb{R}-\{0\}\right\}$ and let $*$ be the matrix multiplication. Determine w…Preview
  10. Q10(i) Let $A$ be $\mathbb{Q}\setminus\{1\}$. Define $*$ on $A$ by $x*y = x+y-xy$. Is $*$ binary on $A$? If so, examine the commutative and ass…Preview
12.3

Mathematical Logic

George Boole (1815-1864), a self-taught English mathematician, philosopher and logician, showed that logic could be studied with the precision of algebra -- his work on Boolean algebra and binary numb…

12.3.1

Statement and Its Truth Value

Communication mostly happens through sentences, which come in several types: declarative (assertive), imperative (a command or request), exclamatory (expressing emotion), interrogative (a question), a…

12.3.2

Compound Statements, Logical Connectives, and Truth Tables

Simple vs compound statements. A simple (atomic) statement cannot be broken into smaller statements. A compound (molecular) statement is built from two or more simple statements.

12.3.3

Tautology, Contradiction, and Contingency

Definition 12.16 (Tautology). A statement is a tautology, denoted , if its truth value is always , no matter the truth values of its component statements.

12.3.4

Duality

Definition 12.19 (Dual). The dual of a statement formula is obtained by replacing every with , every with , every with , and every with .

12.3.5

Logical Equivalence

Definition 12.20. Two compound statements are logically equivalent (or simply equivalent), written or , if the columns for and in a shared truth table are identical in every row.

+Exercise 12.2i15 questions
  1. Q1Let $p$: Jupiter is a planet and $q$: India is an island be any two simple statements. Give verbal sentence describing each of the following…Free
  2. Q2Write each of the following sentences in symbolic form using statement variables $p$ and $q$. (i) 19 is not a prime number and all the angle…Free
  3. Q3Determine the truth value of each of the following statements. (i) If $6+2=5$, then the milk is white. (ii) China is in Europe or $\sqrt3$ i…Free
  4. Q4Which one of the following sentences is a proposition? (i) $4+7=12$ (ii) What are you doing? (iii) $3^n \le 81,\ n\in\mathbb{N}$ (iv) Peacoc…Preview
  5. Q5Write the converse, inverse, and contrapositive of each of the following implication. (i) If $x$ and $y$ are numbers such that $x=y$, then $…Preview
  6. Q6Construct the truth table for the following statements. (i) $\neg p\wedge\neg q$ (ii) $\neg(p\wedge\neg q)$ (iii) $(p\vee q)\vee\neg q$ (iv)…Preview
  7. Q7Verify whether the following compound propositions are tautologies or contradictions or contingency. (i) $(p\wedge q)\wedge\neg(p\vee q)$ (i…Preview
  8. Q8Show that (i) $\neg(p\wedge q)\equiv \neg p\vee\neg q$ (ii) $\neg(p\to q)\equiv p\wedge\neg q$.Preview
  9. Q9Prove that $q\to p \equiv \neg p\to\neg q$.Preview
  10. Q10Show that $p\to q$ and $q\to p$ are not equivalent.Preview
  11. Q11Show that $\neg(p\leftrightarrow q)\equiv p\leftrightarrow \neg q$.Preview
  12. Q12Check whether the statement $p\to(q\to p)$ is a tautology or a contradiction without using the truth table.Preview
  13. Q13Using truth table check whether the statements $\neg(p\vee q)\vee(\neg p\wedge q)$ and $\neg p$ are logically equivalent.Preview
  14. Q14Prove $p\to(q\to r)\equiv (p\wedge q)\to r$ without using truth table.Preview
  15. Q15Prove that $p\to(\neg q\vee r)\equiv \neg p\vee(\neg q\vee r)$ using truth table.Preview
12.4

Choose the Correct or the Most Suitable Answer

This closing 20-question multiple-choice set (Exercise 12.3) draws on every idea across both halves of the chapter.

+Exercise 12.3i20 questions
  1. Q1A binary operation on a set $S$ is a function from (1) $S\to S$ (2) $(S\times S)\to S$ (3) $S\to(S\times S)$ (4) $(S\times S)\to(S\times S)$Free
  2. Q2Subtraction is not a binary operation in (1) $\mathbb{R}$ (2) $\mathbb{Z}$ (3) $\mathbb{N}$ (4) $\mathbb{Q}$Free
  3. Q3Which one of the following is a binary operation on $\mathbb{N}$? (1) Subtraction (2) Multiplication (3) Division (4) All the aboveFree
  4. Q4In the set $\mathbb{R}$ of real numbers '$*$' is defined as follows. Which one of the following is not a binary operation on $\mathbb{R}$? (…Preview
  5. Q5The operation $*$ defined by $a*b=\dfrac{ab}{7}$ is not a binary operation on (1) $\mathbb{Q}^+$ (2) $\mathbb{Z}$ (3) $\mathbb{R}$ (4) $\mat…Preview
  6. Q6In the set $\mathbb{Q}$ define $a\odot b = a+b+ab$. For what value of $y$, $3\odot(y\odot 5)=7$? (1) $y=\dfrac23$ (2) $y=\dfrac{-2}3$ (3) $y…Preview
  7. Q7If $a*b=\sqrt{a^2+b^2}$ on the real numbers then $*$ is (1) commutative but not associative (2) associative but not commutative (3) both com…Preview
  8. Q8Which one of the following statements has the truth value $T$? (1) $\sin x$ is an even function. (2) Every square matrix is non-singular. (3…Preview
  9. Q9Which one of the following statements has truth value $F$? (1) Chennai is in India or $\sqrt2$ is an integer (2) Chennai is in India or $\sq…Preview
  10. Q10If a compound statement involves 3 simple statements, then the number of rows in the truth table is (1) 9 (2) 8 (3) 6 (4) 3Preview
  11. Q11Which one is the inverse of the statement $(p\vee q)\to(p\wedge q)$? (1) $(p\wedge q)\to(p\vee q)$ (2) $\neg(p\vee q)\to(p\wedge q)$ (3) $(\…Preview
  12. Q12Which one is the contrapositive of the statement $(p\vee q)\to r$? (1) $\neg r\to(\neg p\wedge\neg q)$ (2) $\neg r\to(p\vee q)$ (3) $r\to(p\…Preview
  13. Q13The truth table for $(p\wedge q)\vee\neg q$ is given below. | $p$ | $q$ | $(p\wedge q)\vee(\neg q)$ | |---|---|---| | T | T | (a) | | T | F…Preview
  14. Q14In the last column of the truth table for $\neg(p\vee\neg q)$ the number of final outcomes of the truth value '$F$' are (1) 1 (2) 2 (3) 3 (4…Preview
  15. Q15Which one of the following is incorrect? For any two propositions $p$ and $q$, we have (1) $\neg(p\vee q)\equiv \neg p\wedge\neg q$ (2) $\ne…Preview
  16. Q16The truth table for $(p\wedge q)\to\neg p$ is given below. | $p$ | $q$ | $(p\wedge q)\to\neg p$ | |---|---|---| | T | T | (a) | | T | F | (b…Preview
  17. Q17The dual of $\neg(p\vee q)\vee[p\vee(p\wedge\neg r)]$ is (1) $\neg(p\wedge q)\wedge[p\vee(p\wedge\neg r)]$ (2) $(p\wedge q)\wedge[p\wedge(p\…Preview
  18. Q18The proposition $p\wedge(\neg p\vee q)$ is (1) a tautology (2) a contradiction (3) logically equivalent to $p\wedge q$ (4) logically equival…Preview
  19. Q19Determine the truth value of each of the following statements: (a) $4+2=5$ and $6+3=9$ (b) $3+2=5$ and $6+1=7$ (c) $4+5=9$ and $1+2=4$ (d) $…Preview
  20. Q20Which one of the following is not true? (1) Negation of a negation of a statement is the statement itself. (2) If the last column of the tru…Preview
12.5

Summary

Binary operations. A binary operation on a non-empty set assigns a unique element to every ordered pair -- so every binary operation automatically satisfies closure.

Sample & Board Papers

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

+Show 39 questions39 questions
  1. Q1The set of positive even integers, with usual addition forms : (a) a finite group (b) only a semi group (c) only a monoid (d) an infinite gr…Preview
  2. Q2Which of the following is a contradiction ? (a) $p\vee q$ (b) $p\wedge q$ (c) $p\vee(\sim p)$ (d) $p\wedge(\sim p)$Preview
  3. Q3Which of the following are statements ? (i) May God bless you. (ii) Rose is a flower. (iii) Milk is white (iv) 1 is a prime number. (a) (i),…Preview
  4. Q4The order of $-i$ in the multiplicative group of $4^{th}$ roots of unity is : (a) $4$ (b) $3$ (c) $2$ (d) $1$Preview
  5. Q5Show that $[(\sim q)\wedge p]\wedge q$ is a contradiction.Preview
  6. Q6If every element of a group is its own inverse then prove that the group is abelian.Preview
  7. Q7In the multiplicative group of $n^{th}$ roots of unity, the inverse of $\omega^k$ is $(k < n)$ : (a) $\omega^{\frac{1}{k}}$ (b) $\omega^{-1}…Preview
  8. Q8If p's truth value is T and q's truth value is F, then which of the following have the truth value T ? (i) $p \vee q$ (ii) $\sim p \vee q$ (…Preview
  9. Q9The order of $[7]$ in $(Z_9, +_9)$ is : (a) 9 (b) 6 (c) 3 (d) 1Preview
  10. Q10In congruence modulo 5, $\{x \in Z / x = 5k+2,\ k \in Z\}$ represents : (a) $[0]$ (b) $[5]$ (c) $[7]$ (d) $[2]$Preview
  11. Q11Verify whether the statement $q \vee [p \vee (\sim q)]$ is a tautology or a contradiction.Preview
  12. Q12Construct the truth table for $(p \wedge q) \vee (\sim r)$.Preview
  13. Q13Show that $\left\{\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}, \begin{pmatrix}\omega & 0\\0 & \omega^2\end{pmatrix}, \begin{pmatrix}\omega^2 &…Preview
  14. Q14If p is T and q is F, then which of the following have the truth value T ? (i) $p \vee q$ (ii) $\sim p \vee q$ (iii) $p \vee \sim q$ (iv) $p…Preview
  15. Q15Which of the following is not true ? (a) If the last column of its truth table contains only F then it is a contradiction (b) Negation of a…Preview
  16. Q16In the set of integers under the operation $*$ defined by $a * b = a + b - 1$, the identity element is : (a) $a$ (b) $0$ (c) $b$ (d) $1$Preview
  17. Q17Which of the following is not a group ? (a) $(Z, \cdot)$ (b) $(Z_n, +_n)$ (c) $(R, +)$ (d) $(Z, +)$Preview
  18. Q18Show that $p \to q$ and $q \to p$ are not equivalent.Preview
  19. Q19(i) Prove that the identity element of a group is unique. (ii) Prove that $(a^{-1})^{-1} = a$ for every $a \in G$, a group.Preview
  20. Q20In the multiplicative group of cube root of unity, the order of $\omega^2$ is : [$\omega$ is a complex cube root of unity] (a) $2$ (b) $1$ (…Preview
  21. Q21Which of the following is a tautology ? (a) $p \vee (\sim p)$ (b) $p \wedge (\sim p)$ (c) $p \vee q$ (d) $p \wedge q$Preview
  22. Q22Show that the set of all non-zero rational numbers is not closed under addition.Preview
  23. Q23Show that $(p \wedge q) \to (p \vee q)$ is a tautology.Preview
  24. Q24(a) State all the five properties of groups. **OR** (b) Prove that the solution of the differential equation: $(5D^2-8D-4)y = 5e^{\frac{-2}{…Preview
  25. Q25Subtraction is not a binary operation in : (a) $\mathbb{Q}$ (b) $\mathbb{R}$ (c) $\mathbb{Z}$ (d) $\mathbb{N}$Preview
  26. Q26Prove that the identity element is unique if it exists.Preview
  27. Q27Prove that $p \to q \equiv \lnot p \vee q$.Preview
  28. Q28Which one of the following is a binary operation on N ? (a) Multiplication (b) Division (c) Subtraction (d) All the abovePreview
  29. Q29Let $*$ be defined on $\mathbb{R}$ by $(a*b)=a+b+ab-7$. Is $*$ binary on $\mathbb{R}$ ? If so, find $3*\left(\dfrac{-7}{15}\right)$.Preview
  30. Q30The operation $*$ defined by $a*b=\dfrac{ab}{7}$ is not a binary operation on : (a) $\mathbb{R}$ (b) $\mathbb{Q}^+$ (c) $\mathbb{C}$ (d) $\m…Preview
  31. Q31Show that $p\to q$ and $q\to p$ are not equivalent.Preview
  32. Q32(a) Prove that $p\to(\lnot q\vee r)\equiv \lnot p\vee(\lnot q\vee r)$ using truth table. **OR** (b) Suppose a person deposits ₹ 10,000 in a…Preview
  33. Q33The number of rows in the truth table of $(p\vee q)\wedge(p\vee r)$ is : (a) $6$ (b) $9$ (c) $3$ (d) $8$Preview
  34. Q34(a) Show that $p\leftrightarrow q\equiv((\sim p)\vee q)\wedge((\sim q)\vee p)$ **OR** (b) Solve the system of linear equations by Cramer's R…Preview
  35. Q35Subtraction is not a binary operation in : (a) $N$ (b) $R$ (c) $Q$ (d) $Z$Preview
  36. Q36Let Q be the set of all Rational numbers. If $*$ is a binary operation defined on Q as $a*b=a+b-ab+7$ and $\left(\dfrac32\right)*m=\dfrac{87…Preview
  37. Q37The dual of $\lnot(p\vee q)\vee[p\vee(p\wedge\lnot r)]$ is : (a) $\lnot(p\wedge q)\wedge[p\wedge(p\wedge r)]$ (b) $\lnot(p\wedge q)\wedge[p\…Preview
  38. Q38Verify (i) Closure property (ii) Associative property and (iii) Existence of identity for the following operation on the given set : $m*n=m+…Preview
  39. Q39(a) Prove that $p\to(\lnot q\vee r)\equiv\lnot p\vee(\lnot q\vee r)$ using truth table. **OR** (b) Prove that $\displaystyle\int_{0}^{\frac{…Preview