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Mathematics · Textbook solutions

Mathematical Logic

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 266 questions

1.1 Statements and Logical Connectives

66 q

Solved Examples

Worked · 24
  1. Which of the following sentences are statements in logic ? Write down the truth values of the statements.
    1.1 Solved Q.1 (i)
    6×4=256 \times 4 = 25
  2. 1.1 Solved Q.1 (ii)
    x+6=9x + 6 = 9
  3. 1.1 Solved Q.1 (iii)
    What are you doing ?
  4. 1.1 Solved Q.1 (iv)
    The quadratic equation x2−5x+6=0x^2 - 5x + 6 = 0 has 2 real roots.
  5. 1.1 Solved Q.1 (v)
    Please, sit down.
  6. 1.1 Solved Q.1 (vi)
    The Moon revolves around the earth.
  7. 1.1 Solved Q.1 (vii)
    Every real number is a complex number.
  8. 1.1 Solved Q.1 (viii)
    He is honest.
  9. 1.1 Solved Q.1 (ix)
    The square of a prime number is a prime number.
  10. Express the following compound statements symbolically without examining the truth values.
    (i)
    2 is an even number and 25 is a perfect square.
  11. (ii)
    A school is open or there is a holiday.
  12. (iii)
    Delhi is in India but Dhaka is not in Srilanka.
  13. (iv)
    3+8≥123 + 8 \geq 12 if and only if 5×4≤255 \times 4 \leq 25.
  14. Write the truth values of the following statements.
    (i)
    3 is a prime number and 4 is a rational number.
  15. (ii)
    All flowers are red or all cows are black.
  16. (iii)
    If Mumbai is in Maharashtra then Delhi is the capital of India.
  17. (iv)
    Milk is white if and only if the Sun rises in the West.
  18. If statements p, q are true and r, s are false, determine the truth values of the following.
    (i)
    ∼p∧(q∨∼r)\sim p \wedge (q \vee \sim r)
  19. (ii)
    (p∧∼r)∧(∼q∨s)(p \wedge \sim r) \wedge (\sim q \vee s)
  20. (iii)
    ∼(p→q)↔(r∧s)\sim (p \to q) \leftrightarrow (r \wedge s)
  21. (iv)
    (∼p→q)∧(r↔s)(\sim p \to q) \wedge (r \leftrightarrow s)
  22. Write the negations of the following.
    (i)
    Price increases
  23. (ii)
    0!≠10! \neq 1
  24. (iii)
    5+4=95 + 4 = 9

Exercise 1.1

Practice · 42
  1. State which of the following are statements. Justify. In case of statement, state its truth value.
    Q.1 (i)
    5+4=135 + 4 = 13.
  2. Q.1 (ii)
    x−3=14x - 3 = 14.
  3. Q.1 (iii)
    Close the door.
  4. Q.1 (iv)
    Zero is a complex number.
  5. Q.1 (v)
    Please get me breakfast.
  6. Q.1 (vi)
    Congruent triangles are similar.
  7. Q.1 (vii)
    x2=xx^2 = x.
  8. Q.1 (viii)
    A quadratic equation cannot have more than two roots.
  9. Q.1 (ix)
    Do you like Mathematics ?
  10. Q.1 (x)
    The sun sets in the west.
  11. Q.1 (xi)
    All real numbers are whole numbers.
  12. Q.1 (xii)
    Can you speak in Marathi ?
  13. Q.1 (xiii)
    x2−6x−7=0x^2 - 6x - 7 = 0, when x=7x = 7
  14. Q.1 (xiv)
    The sum of cuberoots of unity is zero.
  15. Q.1 (xv)
    It rains heavily.
  16. Write the following compound statements symbolically.
    Q.2 (i)
    Nagpur is in Maharashtra and Chennai is in Tamilnadu
  17. Q.2 (ii)
    Triangle is equilateral or isosceles.
  18. Q.2 (iii)
    The angle is right angle if and only if it is of measure 90∘90^\circ.
  19. Q.2 (iv)
    Angle is neither acute nor obtuse.
  20. Q.2 (v)
    If ΔABC\Delta ABC is right angled at B, then m ∠A\angle A + m ∠C\angle C = 90∘90^\circ
  21. Q.2 (vi)
    Hima Das wins gold medal if and only if she runs fast.
  22. Q.2 (vii)
    xx is not irrational number but is a square of an integer.
  23. Write the truth values of the following.
    Q.3 (i)
    4 is odd or 1 is prime.
  24. Q.3 (ii)
    64 is a perfect square and 46 is a prime number.
  25. Q.3 (iii)
    5 is a prime number and 7 divides 94.
  26. Q.3 (iv)
    It is not true that 5-3i is a real number.
  27. Q.3 (v)
    If 3×5=83 \times 5 = 8 then 3+5=153 + 5 = 15.
  28. Q.3 (vi)
    Milk is white if and only if sky is blue.
  29. Q.3 (vii)
    24 is a composite number or 17 is a prime number.
  30. If the statements p, q are true statements and r, s are false statements then determine the truth values of the following.
    Q.4 (i)
    p∨(q∧r)p \vee (q \wedge r)
  31. Q.4 (ii)
    (p→q)∨(r→s)(p \to q) \vee (r \to s)
  32. Q.4 (iii)
    (q∧r)∨(∼p∧s)(q \wedge r) \vee (\sim p \wedge s)
  33. Q.4 (iv)
    (p→q)∧∼r(p \to q) \wedge \sim r
  34. Q.4 (v)
    (∼r↔p)→∼q(\sim r \leftrightarrow p) \to \sim q
  35. Q.4 (vi)
    [∼p∧(∼q∧r)]∨[(q∧r)∨(p∧r)][\sim p \wedge (\sim q \wedge r)] \vee [(q \wedge r) \vee (p \wedge r)]
  36. Q.4 (vii)
    [(∼p∧q)∧∼r]∨[(q→p)→(∼s∨r)][(\sim p \wedge q) \wedge \sim r] \vee [(q \to p) \to (\sim s \vee r)]
  37. Q.4 (viii)
    ∼[(∼p∧r)∨(s→∼q)]↔(p∧r)\sim [(\sim p \wedge r) \vee (s \to \sim q)] \leftrightarrow (p \wedge r)
  38. Write the negations of the following.
    Q.5 (i)
    Tirupati is in Andhra Pradesh
  39. Q.5 (ii)
    3 is not a root of the equation x2+3x−18=0x^2 + 3x - 18 = 0.
  40. Q.5 (iii)
    2\sqrt{2} is a rational number.
  41. Q.5 (iv)
    Polygon ABCDE is a pentagon.
  42. Q.5 (v)
    7+3>57 + 3 > 5

1.2 Statement Pattern, Logical Equivalence, Tautology, Contradiction and Contingency

31 q

Solved Examples

Worked · 3
  1. 1.2 Solved Ex.1
    Construct the truth table for each of the following statement patterns. i) p→(q→p)p \to (q \to p) ii) (∼p∨q)↔∼(p∧q)(\sim p \vee q) \leftrightarrow \sim (p \wedge q) iii) ∼(∼p∧∼q)∨q\sim (\sim p \wedge \sim q) \vee q iv) [(p∧q)∨r]∧[∼r∨(p∧q)][(p \wedge q) \vee r] \wedge [\sim r \vee (p \wedge q)] v) [(∼p∨q)∧(q→r)]→(p→r)[(\sim p \vee q) \wedge (q \to r)] \to (p \to r)
  2. 1.2 Solved Ex.2
    Using truth tables, prove the following logical equivalences. i) (p∧q)≡∼(p→∼q)(p \wedge q) \equiv \sim (p \to \sim q) ii) (p↔q)≡(p∧q)∨(∼p∧∼q)(p \leftrightarrow q) \equiv (p \wedge q) \vee (\sim p \wedge \sim q) iii) (p∧q)→r≡p→(q→r)(p \wedge q) \to r \equiv p \to (q \to r) iv) p→(q∨r)≡(p→q)∨(p→r)p \to (q \vee r) \equiv (p \to q) \vee (p \to r)
  3. 1.2 Solved Ex.3
    Using truth tables, examine whether each of the following statement pattern is a tautology or a contradiction or contingency. i) (p∧q)∧(∼p∨∼q)(p \wedge q) \wedge (\sim p \vee \sim q) ii) [p∧(p→∼q)]→q[p \wedge (p \to \sim q)] \to q iii) (p→q)∧[(q→r)→(p→r)](p \to q) \wedge [(q \to r) \to (p \to r)] iv) [(p∨q)∨r]↔[p∨(q∨r)][(p \vee q) \vee r] \leftrightarrow [p \vee (q \vee r)]

Exercise 1.2

Practice · 28
  1. Construct the truth table for each of the following statement patterns.
    Q.1 (i)
    [(p→q)∧q]→p[(p \to q) \wedge q] \to p
  2. Q.1 (ii)
    (p∧∼q)↔(p→q)(p \wedge \sim q) \leftrightarrow (p \to q)
  3. Q.1 (iii)
    (p∧q)↔(q∨r)(p \wedge q) \leftrightarrow (q \vee r)
  4. Q.1 (iv)
    p→[∼(q∧r)]p \to [\sim (q \wedge r)]
  5. Q.1 (v)
    ∼p∧[(p∨∼q)∧q]\sim p \wedge [(p \vee \sim q) \wedge q]
  6. Q.1 (vi)
    (∼p→∼q)∧(∼q→∼p)(\sim p \to \sim q) \wedge (\sim q \to \sim p)
  7. Q.1 (vii)
    (q→p)∨(∼p↔q)(q \to p) \vee (\sim p \leftrightarrow q)
  8. Q.1 (viii)
    [p→(q→r)]↔[(p∧q)→r][p \to (q \to r)] \leftrightarrow [(p \wedge q) \to r]
  9. Q.1 (x)
    (p∨∼q)→(r∧p)(p \vee \sim q) \to (r \wedge p)
  10. Using truth tables prove the following logical equivalences.
    Q.2 (i)
    ∼p∧q≡(p∨q)∧∼p\sim p \wedge q \equiv (p \vee q) \wedge \sim p
  11. Q.2 (ii)
    ∼(p∨q)∨(∼p∧q)≡∼p\sim (p \vee q) \vee (\sim p \wedge q) \equiv \sim p
  12. Q.2 (iii)
    p↔q≡∼[(p∨q)∧∼(p∧q)]p \leftrightarrow q \equiv \sim [(p \vee q) \wedge \sim (p \wedge q)]
  13. Q.2 (iv)
    p→(q→p)≡∼p→(p→q)p \to (q \to p) \equiv \sim p \to (p \to q)
  14. Q.2 (v)
    (p∨q)→r≡(p→r)∧(q→r)(p \vee q) \to r \equiv (p \to r) \wedge (q \to r)
  15. Q.2 (vi)
    p→(q∧r)≡(p→q)∧(p→r)p \to (q \wedge r) \equiv (p \to q) \wedge (p \to r)
  16. Q.2 (vii)
    p∧(q∨r)≡(p∧q)∨(p∧r)p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)
  17. Q.2 (viii)
    [∼(p∨q)∨(p∨q)]∧r≡r[\sim (p \vee q) \vee (p \vee q)] \wedge r \equiv r
  18. Q.2 (ix)
    ∼(p↔q)≡(p∧∼q)∨(q∧∼p)\sim (p \leftrightarrow q) \equiv (p \wedge \sim q) \vee (q \wedge \sim p)
  19. Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency.
    Q.3 (i)
    (p∧q)→(q∨p)(p \wedge q) \to (q \vee p)
  20. Q.3 (ii)
    (p→q)↔(∼p∨q)(p \to q) \leftrightarrow (\sim p \vee q)
  21. Q.3 (iii)
    [∼(∼p∧∼q)]∨q[\sim (\sim p \wedge \sim q)] \vee q
  22. Q.3 (iv)
    [(p→q)∧q]→p[(p \to q) \wedge q] \to p
  23. Q.3 (v)
    [(p→q)∧∼q]→∼p[(p \to q) \wedge \sim q] \to \sim p
  24. Q.3 (vi)
    (p↔q)∧(p→∼q)(p \leftrightarrow q) \wedge (p \to \sim q)
  25. Q.3 (vii)
    ∼(∼q∧p)∧q\sim (\sim q \wedge p) \wedge q
  26. Q.3 (viii)
    (p∧∼q)↔(p→q)(p \wedge \sim q) \leftrightarrow (p \to q)
  27. Q.3 (ix)
    (∼p→q)∧(p∧r)(\sim p \to q) \wedge (p \wedge r)
  28. Q.3 (x)
    [p→(∼q∨r)]↔∼[p→(q→r)][p \to (\sim q \vee r)] \leftrightarrow \sim [p \to (q \to r)]

1.3 Quantifiers, Duality, Negation, Converse, Inverse and Contrapositive

49 q

Solved Examples

Worked · 21
  1. If A = {1, 2, 3, 4, 5, 6, 7}, determine the truth value of the following.
    (i)
    ∃x∈A\exists x \in A such that x−4=3x - 4 = 3
  2. (ii)
    ∀x∈A\forall x \in A, x+1≥3x + 1 \geq 3
  3. (iii)
    ∀x∈A\forall x \in A, 8−x≤78 - x \leq 7
  4. (iv)
    ∃x∈A\exists x \in A, such that x+8=16x + 8 = 16
  5. Write the duals of each of the following.
    Duals (i)
    (p∧q)∨r(p \wedge q) \vee r
  6. Duals (ii)
    t∨(p∨q)t \vee (p \vee q)
  7. Duals (iii)
    p∧[∼q∨(p∧q)∨∼r]p \wedge [\sim q \vee (p \wedge q) \vee \sim r]
  8. Duals (iv)
    (p∨q)∧t(p \vee q) \wedge t
  9. Duals (v)
    (p∨q)∨r≡p∨(q∨r)(p \vee q) \vee r \equiv p \vee (q \vee r)
  10. Duals (vi)
    p∧q∧rp \wedge q \wedge r
  11. Duals (vii)
    (p∧t)∨(c∧∼q)(p \wedge t) \vee (c \wedge \sim q)
  12. Write the negations of the following.
    Negations (i)
    3+3<53 + 3 < 5 or 5+5=95 + 5 = 9
  13. Negations (ii)
    7>37 > 3 and 4>114 > 11
  14. Negations (iii)
    The number is neither odd nor perfect square.
  15. Negations (iv)
    The number is an even number if and only if it is divisible by 2.
  16. Write the negations of the following statements.
    (i)
    All natural numbers are rational.
  17. (ii)
    Some students of class X are sixteen year old.
  18. (iii)
    ∃n∈N\exists n \in N such that n+8>11n + 8 > 11
  19. (iv)
    ∀x∈N\forall x \in N, 2x+12x + 1 is odd
  20. Write the converse, inverse and contrapositive of the following statements.
    (i)
    If a function is differentiable then it is continuous.
  21. (ii)
    If it rains then the match will be cancelled.

Exercise 1.3

Practice · 28
  1. If A = {3, 5, 7, 9, 11, 12}, determine the truth value of each of the following.
    Q.1 (i)
    ∃x∈A\exists x \in A such that x−8=1x - 8 = 1
  2. Q.1 (ii)
    ∀x∈A\forall x \in A, x2+xx^2 + x is an even number
  3. Q.1 (iii)
    ∃x∈A\exists x \in A such that x2<0x^2 < 0
  4. Q.1 (iv)
    ∀x∈A\forall x \in A, xx is an even number
  5. Q.1 (v)
    ∃x∈A\exists x \in A such that 3x+8>403x + 8 > 40
  6. Q.1 (vi)
    ∀x∈A\forall x \in A, 2x+9>142x + 9 > 14
  7. Write the duals of each of the following.
    Q.2 (i)
    p∨(q∧r)p \vee (q \wedge r)
  8. Q.2 (ii)
    p∧(q∧r)p \wedge (q \wedge r)
  9. Q.2 (iii)
    (p∨q)∧(r∨s)(p \vee q) \wedge (r \vee s)
  10. Q.2 (iv)
    p∧∼qp \wedge \sim q
  11. Q.2 (v)
    (∼p∨q)∧(∼r∧s)(\sim p \vee q) \wedge (\sim r \wedge s)
  12. Q.2 (vi)
    ∼p∧(∼q∧(p∨q)∧∼r)\sim p \wedge (\sim q \wedge (p \vee q) \wedge \sim r)
  13. Q.2 (vii)
    [∼(p∨q)]∧[p∨∼(q∧∼s)][\sim (p \vee q)] \wedge [p \vee \sim (q \wedge \sim s)]
  14. Q.2 (viii)
    c∨{p∧(q∨r)}c \vee \{p \wedge (q \vee r)\}
  15. Q.2 (ix)
    ∼p∨(q∧r)∧t\sim p \vee (q \wedge r) \wedge t
  16. Q.2 (x)
    (p∨q)∨c(p \vee q) \vee c
  17. Write the negations of the following.
    Q.3 (i)
    x+8>11x + 8 > 11 or y−3=6y - 3 = 6
  18. Q.3 (ii)
    11<1511 < 15 and 25>2025 > 20
  19. Q.3 (iii)
    Qudrilateral is a square if and only if it is a rhombus.
  20. Q.3 (iv)
    It is cold and raining.
  21. Q.3 (v)
    If it is raining then we will go and play football.
  22. Q.3 (vii)
    All natural numbers are whole numbers.
  23. Q.3 (viii)
    ∀n∈N\forall n \in N, n2+n+2n^2 + n + 2 is divisible by 4.
  24. Q.3 (ix)
    ∃x∈N\exists x \in N such that x−17<20x - 17 < 20
  25. Write converse, inverse and contrapositive of the following statements.
    Q.4 (i)
    If x<yx < y then x2<y2x^2 < y^2 (x,y∈R)(x, y \in R)
  26. Q.4 (ii)
    A family becomes literate if the woman in it is literate.
  27. Q.4 (iii)
    If surface area decreases then pressure increases.
  28. Q.4 (iv)
    If voltage increases then current decreases.

1.4 Some Important Results (Algebra of Statements)

25 q

Solved Examples

Worked · 10
  1. Write the negations of the following stating the rules used.
    (i)
    (p∨q)∧(q∨∼r)(p \vee q) \wedge (q \vee \sim r)
  2. (ii)
    (p→q)∨r(p \to q) \vee r
  3. (iii)
    p∧(q∨r)p \wedge (q \vee r)
  4. (iv)
    (∼p∧q)∨(p∧∼q)(\sim p \wedge q) \vee (p \wedge \sim q)
  5. (v)
    (p∧q)→(∼p∨r)(p \wedge q) \to (\sim p \vee r)
  6. Rewrite the following statements without using if ...... then.
    (i)
    If prices increase then the wages rise.
  7. (ii)
    If it is cold, then we wear woolen clothes.
  8. Without using truth table prove that :
    (i)
    p↔q≡∼(p∧∼q)∧∼(q∧∼p)p \leftrightarrow q \equiv \sim (p \wedge \sim q) \wedge \sim (q \wedge \sim p)
  9. (ii)
    ∼(p∨q)∨(∼p∧q)≡∼p\sim (p \vee q) \vee (\sim p \wedge q) \equiv \sim p
  10. (iii)
    ∼p∧q≡(p∨q)∧∼p\sim p \wedge q \equiv (p \vee q) \wedge \sim p

Exercise 1.4

Practice · 15
  1. Using rules of negation write the negations of the following with justification.
    Q.1 (i)
    ∼q→p\sim q \to p
  2. Q.1 (ii)
    p∧∼qp \wedge \sim q
  3. Q.1 (iii)
    p∨∼qp \vee \sim q
  4. Q.1 (iv)
    (p∨∼q)∧r(p \vee \sim q) \wedge r
  5. Q.1 (v)
    p→(p∨∼q)p \to (p \vee \sim q)
  6. Q.1 (vi)
    ∼(p∧q)∨(p∨∼q)\sim (p \wedge q) \vee (p \vee \sim q)
  7. Q.1 (vii)
    (p∨∼q)→(p∧∼q)(p \vee \sim q) \to (p \wedge \sim q)
  8. Q.1 (viii)
    (∼p∨∼q)∨(p∧∼q)(\sim p \vee \sim q) \vee (p \wedge \sim q)
  9. Rewrite the following statements without using if .. then.
    Q.2 (i)
    If a man is a judge then he is honest.
  10. Q.2 (ii)
    If 2 is a rational number then 2\sqrt{2} is irrational number.
  11. Q.2 (iii)
    If f(2)=0f(2) = 0 then f(x)f(x) is divisible by (x−2)(x - 2).
  12. Without using truth table prove that :
    Q.3 (i)
    p↔q≡(p∧q)∨(∼p∧∼q)p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)
  13. Q.3 (ii)
    (p∨q)∧(p∨∼q)≡p(p \vee q) \wedge (p \vee \sim q) \equiv p
  14. Q.3 (iii)
    (p∧q)∨(∼p∧q)∨(p∧∼q)≡p∨q(p \wedge q) \vee (\sim p \wedge q) \vee (p \wedge \sim q) \equiv p \vee q
  15. Q.3 (iv)
    ∼[(p∨∼q)→(p∧∼q)]≡(p∨∼q)∧(∼p∨q)\sim [(p \vee \sim q) \to (p \wedge \sim q)] \equiv (p \vee \sim q) \wedge (\sim p \vee q)

1.5 Application of Logic to Switching Circuits

31 q

Solved Examples

Worked · 10
  1. Construct switching circuits of the following.
    (i)
    Construct the switching circuit of [(p∨(∼p∧q)]∨[(∼q∧r)∨∼p][(p \vee (\sim p \wedge q)] \vee [(\sim q \wedge r) \vee \sim p].
  2. (ii)
    Construct the switching circuit of (p∧q∧r)∨[∼p∨(q∧∼r)](p \wedge q \wedge r) \vee [\sim p \vee (q \wedge \sim r)].
  3. (iii)
    Construct the switching circuit of [(p∧r)∨(∼q∧∼r)]∨(∼p∧∼r)[(p \wedge r) \vee (\sim q \wedge \sim r)] \vee (\sim p \wedge \sim r).
  4. Q.1. (Solved) Express the following circuits in the symbolic form of logic and write the input-output table.
    (i)
    Circuit (i)
  5. (ii)
    Circuit (ii)
  6. (iii)
    Circuit (iii)
  7. Ex.3. Give an alternative arrangement for the following circuit, so that the new circuit has minimum switches.
    1.5 Solved Ex.3
    Circuit (Fig. 1.11)
  8. Ex.4. Express the following switching circuit in the symbolic form of Logic. Construct the switching table and interpret it.
    1.5 Solved Ex.4
    Circuit (Fig. 1.13)
  9. Ex.5. Simplify the given circuit by writing its logical expression. Also, write your conclusion.
    1.5 Solved Ex.5
    Circuit (Fig. 1.14)
  10. Ex.6. In the following switching circuit, (i) write symbolic form, (ii) construct switching table, (iii) simplify the circuit.
    1.5 Solved Ex.6
    Circuit (Fig. 1.15)

Exercise 1.5

Practice · 21
  1. Construct the switching circuit of the following.
    Q.2 (i)
    Construct the switching circuit of (∼p∧q)∨(p∧∼r)(\sim p \wedge q) \vee (p \wedge \sim r).
  2. Q.2 (ii)
    Construct the switching circuit of (p∧q)∨[∼p∧(∼q∨p∨r)](p \wedge q) \vee [\sim p \wedge (\sim q \vee p \vee r)].
  3. Q.2 (iii)
    Construct the switching circuit of [(p∧r)∨(∼q∧∼r)]∧(∼p∧∼r)[(p \wedge r) \vee (\sim q \wedge \sim r)] \wedge (\sim p \wedge \sim r).
  4. Q.2 (iv)
    Construct the switching circuit of (p∧∼q∧r)∨[p∧(∼q∨∼r)](p \wedge \sim q \wedge r) \vee [p \wedge (\sim q \vee \sim r)].
  5. Q.2 (v)
    Construct the switching circuit of p∨(∼p)∨(∼q)∨(p∧q)p \vee (\sim p) \vee (\sim q) \vee (p \wedge q).
  6. Q.2 (vi)
    Construct the switching circuit of (p∧q)∨(∼p)∨(p∧∼q)(p \wedge q) \vee (\sim p) \vee (p \wedge \sim q).
  7. Obtain the simple logical expression of the following. Draw the corresponding switching circuit.
    Q.5 (i)
    Obtain the simple logical expression of p∨(q∧∼q)p \vee (q \wedge \sim q) and draw the corresponding switching circuit.
  8. Q.5 (ii)
    Obtain the simple logical expression of (∼p∧q)∨(∼p∧∼q)∨(p∧∼q)(\sim p \wedge q) \vee (\sim p \wedge \sim q) \vee (p \wedge \sim q) and draw the corresponding switching circuit.
  9. Q.5 (iii)
    Obtain the simple logical expression of [p∨(∼q)∨∼r)]∧(p∨(q∧r))[p \vee (\sim q) \vee \sim r)] \wedge (p \vee (q \wedge r)) and draw the corresponding switching circuit.
  10. Q.5 (iv)
    Obtain the simple logical expression of (p∧q∧∼p)∨(∼p∧q∧r)∨(p∧∼q∧r)∨(p∧q∧r)(p \wedge q \wedge \sim p) \vee (\sim p \wedge q \wedge r) \vee (p \wedge \sim q \wedge r) \vee (p \wedge q \wedge r) and draw the corresponding switching circuit.
  11. Q.1. Express the following circuits in the symbolic form of logic and write the input-output table.
    Q.1 (i)
    Circuit (i)
  12. Q.1 (ii)
    Circuit (ii)
  13. Q.1 (iii)
    Circuit (iii)
  14. Q.1 (iv)
    Circuit (iv)
  15. Q.1 (v)
    Circuit (v)
  16. Q.1 (vi)
    Circuit (vi)
  17. Q.3. Give an alternative equivalent simple circuits for the following circuits.
    Q.3 (i)
    Circuit (i)
  18. Q.3 (ii)
    Circuit (ii)
  19. Q.4. Write the symbolic form of the following switching circuits, construct its switching table and interpret it.
    Q.4 (i)
    Circuit (i)
  20. Q.4 (ii)
    Circuit (ii)
  21. Q.4 (iii)
    Circuit (iii)

Miscellaneous Exercise 1

64 q

Choose the correct alternative

Practice · 7
  1. Misc I (i)
    If p∧qp \wedge q is false and p∨qp \vee q is true, the ________ is not true.

    Tap an option to check your answer.

  2. Misc I (ii)
    (p∧q)→r(p \wedge q) \to r is logically equivalent to ________.

    Tap an option to check your answer.

  3. Misc I (iii)
    Inverse of statement pattern (p∨q)→(p∧q)(p \vee q) \to (p \wedge q) is ________.

    Tap an option to check your answer.

  4. Misc I (iv)
    If p∧qp \wedge q is F, p→qp \to q is F then the truth values of pp and qq are ________.

    Tap an option to check your answer.

  5. Misc I (v)
    The negation of inverse of ∼p→q\sim p \to q is ________.

    Tap an option to check your answer.

  6. Misc I (vi)
    The negation of p∧(q→r)p \wedge (q \to r) is ________.

    Tap an option to check your answer.

  7. Misc I (vii)
    If A={1,2,3,4,5}A = \{1, 2, 3, 4, 5\} then which of the following is not true?

    Tap an option to check your answer.

Solve the following

Practice · 57
  1. Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements:
    Misc Q.2 (i)
    4!=244! = 24.
  2. Misc Q.2 (ii)
    π\pi is an irrational number.
  3. Misc Q.2 (iii)
    India is a country and Himalayas is a river.
  4. Misc Q.2 (iv)
    Please get me a glass of water.
  5. Misc Q.2 (v)
    cos⁡2θ−sin⁡2θ=cos⁡2θ\cos^2\theta - \sin^2\theta = \cos 2\theta for all θ∈R\theta \in R.
  6. Misc Q.2 (vi)
    If xx is a whole number the x+6=0x + 6 = 0.
  7. Write the truth values of the following statements:
    Misc Q.3 (i)
    5\sqrt{5} is an irrational but 353\sqrt{5} is a complex number.
  8. Misc Q.3 (ii)
    ∀n∈N\forall n \in N, n2+nn^2 + n is even number while n2−nn^2 - n is an odd number.
  9. Misc Q.3 (iii)
    ∃n∈N\exists n \in N such that n+5>10n + 5 > 10.
  10. Misc Q.3 (iv)
    The square of any even number is odd or the cube of any odd number is odd.
  11. Misc Q.3 (v)
    In △ABC\triangle ABC if all sides are equal then its all angles are equal.
  12. Misc Q.3 (vi)
    ∀n∈N\forall n \in N, n+6>8n + 6 > 8.
  13. If A={1,2,3,4,5,6,7,8,9}A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}, determine the truth value of each of the following statement:
    Misc Q.4 (i)
    ∃x∈A\exists x \in A such that x+8=15x + 8 = 15.
  14. Misc Q.4 (ii)
    ∀x∈A\forall x \in A, x+5<12x + 5 < 12.
  15. Misc Q.4 (iii)
    ∃x∈A\exists x \in A, such that x+7≥11x + 7 \geq 11.
  16. Misc Q.4 (iv)
    ∀x∈A\forall x \in A, 3x≤253x \leq 25.
  17. Write the negations of the following:
    Misc Q.5 (i)
    ∀n∈A\forall n \in A, n+7>6n + 7 > 6.
  18. Misc Q.5 (ii)
    ∃x∈A\exists x \in A, such that x+9≤15x + 9 \leq 15.
  19. Misc Q.5 (iii)
    Some triangles are equilateral triangle.
  20. Construct the truth table for each of the following:
    Misc Q.6 (i)
    p→(q→p)p \to (q \to p)
  21. Misc Q.6 (ii)
    (∼p∨∼q)↔[∼(p∧q)](\sim p \vee \sim q) \leftrightarrow [\sim(p \wedge q)]
  22. Misc Q.6 (iii)
    ∼(∼p∧∼q)∨q\sim(\sim p \wedge \sim q) \vee q
  23. Misc Q.6 (iv)
    [(p∧q)∨r]∧[∼r∨(p∧q)][(p \wedge q) \vee r] \wedge [\sim r \vee (p \wedge q)]
  24. Misc Q.6 (v)
    [(∼p∨q)∧(q→r)]→(p→r)[(\sim p \vee q) \wedge (q \to r)] \to (p \to r)
  25. Determine whether the following statement patterns are tautologies contradictions or contingencies:
    Misc Q.7 (i)
    [(p→q)∧∼q)]→∼p[(p \to q) \wedge \sim q)] \to \sim p
  26. Misc Q.7 (ii)
    [(p∨q)∧∼p]∧∼q[(p \vee q) \wedge \sim p] \wedge \sim q
  27. Misc Q.7 (iii)
    (p→q)∧(p∧∼q)(p \to q) \wedge (p \wedge \sim q)
  28. Misc Q.7 (iv)
    [p→(q→r)]↔[(p∧q)→r][p \to (q \to r)] \leftrightarrow [(p \wedge q) \to r]
  29. Misc Q.7 (v)
    [p∧(p→q)]→q[p \wedge (p \to q)] \to q
  30. Misc Q.7 (vi)
    (p∧q)∨(∼p∧q)∨(p∨∼q)∨(∼p∧∼q)(p \wedge q) \vee (\sim p \wedge q) \vee (p \vee \sim q) \vee (\sim p \wedge \sim q)
  31. Misc Q.7 (vii)
    [(p∨∼q)∨(∼p∧q)]∧r[(p \vee \sim q) \vee (\sim p \wedge q)] \wedge r
  32. Misc Q.7 (viii)
    (p→q)∨(q→p)(p \to q) \vee (q \to p)
  33. Determine the truth values of pp and qq in the following cases:
    Misc Q.8 (i)
    (p∨q)(p \vee q) is T and (p∧q)(p \wedge q) is T
  34. Misc Q.8 (ii)
    (p∨q)(p \vee q) is T and (p∨q)→q(p \vee q) \to q is F
  35. Misc Q.8 (iii)
    (p∧q)(p \wedge q) is F and (p∧q)→q(p \wedge q) \to q is T
  36. Using truth tables prove the following logical equivalences:
    Misc Q.9 (i)
    p↔q≡(p∧q)∨(∼p∧∼q)p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)
  37. Misc Q.9 (ii)
    (p∧q)→r≡p→(q→r)(p \wedge q) \to r \equiv p \to (q \to r)
  38. Using rules in logic, prove the following:
    Misc Q.10 (i)
    p↔q≡∼(p∧∼q)∧∼(q∧∼p)p \leftrightarrow q \equiv \sim(p \wedge \sim q) \wedge \sim(q \wedge \sim p)
  39. Misc Q.10 (ii)
    ∼p∧q≡(p∨q)∧∼p\sim p \wedge q \equiv (p \vee q) \wedge \sim p
  40. Misc Q.10 (iii)
    ∼(p∨q)∨(∼p∧q)≡∼p\sim(p \vee q) \vee (\sim p \wedge q) \equiv \sim p
  41. Using the rules in logic, write the negations of the following:
    Misc Q.11 (i)
    (p∨q)∧(q∨∼r)(p \vee q) \wedge (q \vee \sim r)
  42. Misc Q.11 (ii)
    p∧(q∨r)p \wedge (q \vee r)
  43. Misc Q.11 (iii)
    (p→q)∧r(p \to q) \wedge r
  44. Misc Q.11 (iv)
    (∼p∧q)∨(p∧∼q)(\sim p \wedge q) \vee (p \wedge \sim q)
  45. Check whether the following switching circuits are logically equivalent - Justify.
    Misc Q.14 (A) i)
    (A) Circuit (i): S1S_1 in series with (S2 in parallel with S3)(S_2 \text{ in parallel with } S_3), i.e. S1∧(S2∨S3)S_1 \wedge (S_2 \vee S_3), and Circuit (ii): (S1∧S2)∨(S1∧S3)(S_1 \wedge S_2) \vee (S_1 \wedge S_3) — check whether they are logically equivalent.
  46. Misc Q.14 (B) i)
    (B) Circuit (i): (S1∨S2)∧(S1∨S3)(S_1 \vee S_2) \wedge (S_1 \vee S_3), and Circuit (ii): S1∨(S2∧S3)S_1 \vee (S_2 \wedge S_3) — check whether they are logically equivalent.
  47. Q.12. Express the following circuits in the symbolic form. Prepare the switching table.
    Misc Q.12 (i)
    Circuit (i) (Fig. 1.30)
  48. Misc Q.12 (ii)
    Circuit (ii) (Fig. 1.31)
  49. Q.13. Simplify the following so that the new circuit has minimum number of switches. Also, draw the simplified circuit.
    Misc Q.13 (i)
    Circuit (i) (Fig. 1.32)
  50. Misc Q.13 (ii)
    Circuit (ii) (Fig. 1.33)
  51. Q.14. Check whether the following switching circuits are logically equivalent - Justify.
    Misc Q.14 (A) circuit 1
    (A) circuit i (Fig. 1.34)
  52. Misc Q.14 (A) circuit 2
    (A) circuit ii (Fig. 1.35)
  53. Q.14. Check whether the following switching circuits are logically equivalent - Justify.
    Misc Q.14 (B) circuit 1
    (B) circuit i (Fig. 1.36)
  54. Misc Q.14 (B) circuit 2
    (B) circuit ii (Fig. 1.37)
  55. Q.15. Give alternative arrangement of the following switching circuit, so that the new circuit has minimum switches.
    Misc Q.15
    Circuit (Fig. 1.38)
  56. Q.16. Simplify the following circuit so that the new circuit has minimum number of switches.
    Misc Q.16
    Circuit (Fig. 1.39)
  57. Q.17. Represent the following switching circuit in symbolic form and construct its switching table. Write your conclusion from the switching table.
    Misc Q.17
    Circuit (Fig. 1.40)