9th Class Mathematics Unit 8 Logic Numericals with Solutions

8.1.1.Four options are given against each statement. Encircle the correct option. (i) Which of the following expressions is often related to inductive reasoning? (a) based on repeated experiments (b) if and only if statements (c) Statement is proven by a theorem (d) based on general principles. (ii) Which of the following sentences describe deductive reasoning? (a) general conclusions from a limited number of observations (b) based on repeated experiments (c) based on units of information that are accurate (d) draw conclusion from well-known facts. (iii) Which one of the following statements is true? (a) The set of integers is finite (b) The sum of the interior angles of any quadrilateral is always 180° (c) 22/7 ∉ Q' (d) All isosceles triangles are equilateral triangles. (iv) Which of the following statements is the best to represent the negation of the statement "The stove is burning"? (a) the stove is not burning (b) the stove is dim (c) the stove is turned to low heat (d) it is both burning and not burning. (v) The conjunction of two statements p and q is true when: (a) both p and q are false (b) both p and q are true (c) only q is true (d) only p is true. (vi) A conditional is regarded as false only when: (a) antecedent is true and consequent is false (b) consequent is true and antecedent is false (c) antecedent is true only (d) consequent is false only. (vii) Contrapositive of q→p is (a) q→~p (b) ~q→p (c) ~p→~q (d) ~q→~p. (viii) The statement "Every integer greater than 2 is a sum of two prime numbers" is: (a) theorem (b) conjecture (c) axiom (d) postulates. (ix) The statement "A straight line can be drawn between any two points" is: (a) theorem (b) conjecture (c) axiom (d) logic. (x) The statement "The sum of the interior angle of a triangle is 180°" is: (a) converse (b) theorem (c) axiom (d) conditional.

Result

(i) (a) based on repeated experiments

(ii) (d) draw conclusion from well-known facts

(iii) (c) 227 notin Q'

(iv) (a) the stove is not burning

(v) (b) both p and q are true

(vi) (a) antecedent is true and consequent is false

(vii) (c) sim p to sim q

(viii) (b) conjecture

(ix) (c) axiom

(x) (b) theorem

8.1.2.Write the converse, inverse and contrapositive of the following conditionals: (i) ~p → q (ii) q → p (iii) ~p → ~q (iv) ~q → ~p

Result

(i) Converse q to sim p

(i) Inverse p to sim q

(i) Contrapositive sim q to p

(ii) Converse p to q

(ii) Inverse sim q to sim p

(ii) Contrapositive sim p to sim q

(iii) Converse sim q to sim p

(iii) Inverse p to q

(iii) Contrapositive q to p

(iv) Converse sim p to sim q

(iv) Inverse q to p

(iv) Contrapositive p to q

8.1.3.Write the truth table of the following (i) ~(p∨q)∨(~q) (ii) ~(~q∨~p) (iii) (p∨q)↔(p∧q)

(i) table begin{array}{cc|c|c|c|c} p&q&pvee q&sim(pvee q)&sim q&sim(pvee q)vee(sim q)hline T&T&T&F&F&FT&F&T&F&T&TF&T&T&F&F&FF&F&F&T&T&Tend{array}

Result

(i) final column F, T, F, T

(ii) table begin{array}{cc|c|c|c|c} p&q&sim p&sim q&(sim qveesim p)&sim(sim qveesim p)hline T&T&F&F&F&TT&F&F&T&T&FF&T&T&F&T&FF&F&T&T&T&Fend{array}

(ii) final column T, F, F, F

(iii) table begin{array}{cc|c|c|c} p&q&(pvee q)&(pwedge q)&(pvee q)leftrightarrow(pwedge q)hline T&T&T&T&TT&F&T&F&FF&T&T&F&FF&F&F&F&Tend{array}

(iii) final column T, F, F, T

8.1.4.Differentiate between a mathematical statement and its proof. Give two examples.

Result

Definitions A statement is a declarative sentence that is true or false; a proof is a logical argument establishing its truth.

Given

True statement examples 227notin Q' ;quad Qsubseteq R

False statement examples 3+4=8 ;quad Zsubseteq W

Proof examples If x is an odd integer, then x² is also an odd integer; the sum of two odd numbers is an even number.

8.1.5.What is the difference between an axiom and a theorem? Give examples of each.

Result

Theorem definition A theorem is a statement proved true from previously known facts.

Given

Theorem examples Sum of interior angles of a quadrilateral = 360^circ; Fundamental Theorem of Arithmetic; Fermat's Last Theorem: aⁿ+bⁿ=cⁿ has no positive-integer solution for n>2

Result

Axiom definition An axiom is a statement accepted as true without proof.

Given

Axiom examples Through a point infinitely many lines can pass; a straight line can be drawn between any two points; every natural number has a successor; two sets are equal iff they have the same elements; every set has a power set.

8.1.6.What is the importance of logical reasoning in mathematical proofs? Give an example to illustrate your point.

Result

Answer Logic lets us interpret statements, examine their truth, and deduce new information; a person generalizing an allergy after one or two penicillin reactions illustrates inductive reasoning from limited observations.

8.1.7.Indicate whether it is an axiom, conjecture or theorem and explain your reasoning. (i) There is exactly one straight line through any two points. (ii) Every even number greater than 2 can be written as the sum of two prime numbers. (iii) The sum of the angles in a triangle is 180°.

Result

(i) Euclidean Axiom — believed true without proof.

(ii) Conjecture — Goldbach's Conjecture, not formally proven or disproven.

(iii) Theorem — formally proven using established axioms and definitions of geometry.

8.1.8.Formulate simple deductive proofs for each of the following algebraic expressions, prove that the L.H.S is equal to the R.H.S: (i) prove that (x-4)^2+9 = x^2-8x+25 (ii) prove that (x+1)^2-(x-1)^2 = 4x (iii) prove that (x+5)^2-(x-5)^2 = 20x

Working

(i) Expand L.H.S. = (x-4)²+9 = x²-8x+16+9

Result

(i) Result = x²-8x+25 = R.H.S.

Working

(ii) Expand L.H.S. = (x+1)²-(x-1)² = (x²+2x+1)-(x²-2x+1)

Result

(ii) Result = x²+2x+1-x²+2x-1 = 4x = R.H.S.

Working

(iii) Expand L.H.S. = (x+5)²-(x-5)² = (x²+10x+25)-(x²-10x+25)

Result

(iii) Result = x²+10x+25-x²+10x-25 = 20x = R.H.S.

8.1.9.Prove the following by justifying each step: (i) (4+16x)/4 = 1+4x (ii) (6x^2+18x)/(3x^2-27) = 2x/(x-3) (iii) (x^2+7x+10)/(x^2-3x-10) = (x+5)/(x-5)

(i) 4+16x4 = 14 × (4 × 1+4 × 4x) = 14 × 4 × (1+4x) = 1 × (1+4x)

Result

(i) Result = 1+4x

(ii) Factorize 6x²+18x3x²-27 = 6x(x+3)3(x²-9) = 2x(x+3)(x-3)(x+3)

(ii) Result = 2xx-3

(iii) Factorize x²+7x+10x²-3x-10 = (x+2)(x+5)(x+2)(x-5)

(iii) Result = x+5x-5

8.1.10.Suppose x is an integer. Then x is odd if and only if 9x+4 is odd.

Given

Forward direction Let x = 2k+1 for some integer k

Working

Forward substitution 9x+4 = 9(2k+1)+4 = 18k+9+4 = 18k+13 = 2(9k+6)+1

Result

Forward result Since 9k+6 is an integer, 9x+4 is odd.

Given

Reverse direction Let 9x+4 = 2m+1 for some integer m

Reverse working 9x = 2m-3 implies x = 2m-39

Reverse working (flawed) x = 2m-39 = 2left(m-19right) - 13 quad (since m is odd, m-1 is even)

This step is algebraically incorrect: 2(m-1)/9 - 1/3 = (2m-2)/9 - 3/9 = (2m-5)/9, which is not equal to (2m-3)/9. There is also no justification given for assuming m is odd. A valid proof of this direction is by contraposition: if x is even, x=2k, then 9x+4 = 18k+4 = 2(9k+2), which is even; hence by contrapositive, 9x+4 odd implies x is odd.

Result

Reverse result Since (m-1)/9 is an integer, x is odd.

Conclusion Therefore, if x is an integer, then x is odd if and only if 9x+4 is odd.

8.1.11.Suppose x is an integer. If x is odd, then 7x+5 is even.

Given

Given Let x = 2k+1 for some integer k

Working

Substitution 7x+5 = 7(2k+1)+5 = 14k+7+5 = 14k+12 = 2(7k+6)

Result

Result Since 7k+6 is an integer, 2(7k+6) is even. Therefore 7x+5 is even.

8.1.12.Prove the following statements (a) If x is an odd integer, then show that x^2-4x+6 is odd. (b) If x is an even integer then show that x^2+2x+4 is even.

Given

(a) Given Let x = 2k+1 for some integer k

Working

(a) Substitution x²-4x+6 = (2k+1)²-4(2k+1)+6 = 4k²+4k+1-8k-4+6

Result

(a) Result = 4k²-4k+3 = 4k(k-1)+3 implies odd

Given

(b) Given Let x = 2k for some integer k

Working

(b) Substitution x²+2x+4 = (2k)²+2(2k)+4 = 4k²+4k+4 = 4(k²+k+1)

Result

(b) Result Since k²+k+1 is an integer, 4(k²+k+1) is even.

8.1.13.Prove that for any two non-empty sets A and B, (A∩B)' = A'∪B'.

Given

Forward inclusion xin(Acap B)' implies xnotin(Acap B) implies xnotin A or xnotin B implies xin A' or xin B' implies xin A'cup B'

Result

Forward result (Acap B)' subseteq A'cup B' quad ...(i)

Given

Reverse inclusion yin A'cup B' implies yin A' or yin B' implies ynotin A or ynotin B implies ynotin(Acap B) implies yin(Acap B)'

Result

Reverse result A'cup B' subseteq (Acap B)' quad ...(ii)

Conclusion (Acap B)' = A'cup B'

8.1.14.If x and y are positive real numbers and x^2 < y^2 then x < y.

Given

Given x² < y²

Taking square root sqrt{x²} < sqrt{y²}

Result

Result x < y

8.1.15.The sum of the interior angles of a triangle is 180°.

Given

Construction Draw line PQ through A, parallel to BC of triangle ABC

Straight angle at A angle PAB + angle BAC + angle QAC = 180^circ quad ...(1)

Alternate angles (PQ || BC) angle QAC = angle ACB, quad angle PAB = angle CBA

Working

Substitute into (1) angle ACB + angle BAC + angle CBA = 180^circ

Result

Result The sum of the interior angles of a triangle is 180^circ.

8.1.16.If a, b and c are non-zero real numbers, prove that: (a) a/b = c/d ⟺ ad = bc (b) (a/b)·(c/d) = ac/bd (c) a/b + c/b = (a+c)/b

Given

(a) Given ab = cd

(a) Multiply both sides by bd ab × bd = cd × bd implies ad = bc

Result

(a) Result ad = bc

Given

(b) Given ab · cd

(b) Rearrange = a × 1b · c × 1d = ac × 1bd

Result

(b) Result = acbd

Given

(c) Given ab+cb

(c) Combine over common denominator = a × 1b+c × 1b = (a+c) × 1b

Result

(c) Result = a+cb

Unit 8: Logic — Exercise 8 1