Recent questions tagged kenneth-rosen / GATE Overflow for GATE CSE

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1081

Kenneth Rosen Edition 6th Exercise 1.3 Example 19 (Page No. 39)
Can someone please explain the following solution. Ques: Show that ∀x(P(x) ∧ Q(x)) and ∀xP(x) ∧ ∀xQ(x) are logically equivalent (where the same domain is used throughout). This logical equivalence shows that we can distribute a universal quantifier over a ... Q(x)) is true. We can now conclude that ∀x(P(x) ∧ Q(x)) ≡ ∀xP(x) ∧ ∀xQ(x).

Can someone please explain the following solution.Ques: Show that ∀x(P(x) ∧ Q(x)) and ∀xP(x) ∧ ∀xQ(x) are logically equivalent (where the same domain is used th...

Navneet Srivastava

1.6k views

Navneet Srivastava asked Jul 1, 2016

1 votes

1 answer

1082

Kenneth Rosen Edition 6th Exercise 1.3 Example 17 (Page No. 38)
The restriction of a universal quantification is the same as the universal quantification of a conditional statement. For instance, ∀x < 0 (x2 > 0) is another way of expressing ∀x(x < 0 ... whereas existential quantification is same as existential quantification of a conjunction? Please provide proper details. Thank You.

The restriction of a universal quantification is the same as the universal quantificationof a conditional statement. For instance, ∀x < 0 (x2 0) is another way of expr...

Navneet Srivastava

554 views

Navneet Srivastava asked Jul 1, 2016

0 votes

2 answers

1083

Kenneth Rosen Edition 6th Exercise 8.1 Question 6 (Page No. 510)
Find a recurrence relation for the number of strictly increasing sequences of positive integers that have 1 as their first term and n as their last term, where n is a positive integer. That is, sequences a1,a2,a3,a4....ak where a1 = 1, ak = n, and aj < aj+1 for j = 1,2,3,...k-1

Find a recurrence relation for the number of strictly increasing sequences of positive integers that have 1 as their first term and n as their last term, where n is a pos...

dd

1.9k views

dd asked Jun 8, 2016

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0 answers

1084

Kenneth Rosen Edition 6th Exercise 2.3 Question 77 (Page No. 149)
Show that the polynomial f: Z+X Z+ $\rightarrow$ Z+ with f(m,n) = (m+n-2)(m+n-1)/2 + m is one to one and onto .

Show that the polynomial f: Z+X Z+ $\rightarrow$ Z+ with f(m,n) = (m+n-2)(m+n-1)/2 + m is one to one and onto .

khushtak

263 views

khushtak asked Jun 6, 2016

1 votes

0 answers

1085

Kenneth Rosen Edition 6th Exercise 2.3 Question 68 (Page No. 148)
Suppose that F is a function from A to B ,where A and B are finite sets with |A| =|B|. Show that f is one to one if and only if it is onto. I want to know my proof is right or wrong? Proof: by contradiction f[A] is image of function f. ... f[A]| = |B| and not one to one : |B| < |A| (which contradicts |A|=|B|) so f is one to one

Suppose that F is a function from A to B ,where A and B are finite sets with |A| =|B|. Show that f is one to one if and only if it is onto.I want to know my proof is righ...

khushtak

846 views

khushtak asked Jun 6, 2016

0 votes

2 answers

1086

Kenneth Rosen Edition 6th Exercise 2.1 Question 31 (Page No. 120)
31 Explain why A X B X C and ( A X B ) X C are not same.

31 Explain why A X B X C and ( A X B ) X C are not same.

khushtak

3.7k views

khushtak asked Jun 3, 2016

2 votes

1 answer

1087

Kenneth Rosen Edition 6th Exercise 2.2 Question 24 (Page No. 131)
Q-24 Let A,B and C be sets. Show that (A - B) -C = (A - C) - (B - C) and I think ( A - B) -C = (A - C) - B

Q-24 Let A,B and C be sets. Show that (A - B) -C = (A - C) - (B - C)and I think ( A - B) -C = (A - C) - B

khushtak

510 views

khushtak asked Jun 3, 2016

0 votes

1 answer

1088

Kenneth Rosen Edition 6th Exercise 2.1 Question 21c (Page No. 120)
21 c) How many elements does the given set have where a and b are distinct elements? P(P(∅))

21 c) How many elements does the given set have where a and b are distinct elements? P(P(∅))

khushtak

572 views

khushtak asked Jun 2, 2016

1 votes

0 answers

1089

Kenneth Rosen Edition 6th Exercise 1.4 Question 44 (Page No. 62)
Use quantifiers and logical connectives to express the fact that a quadratic polynomial with real number coefficients has at most two real roots. answer I think is ∀a∀b∀c[∃x ∃y ((ax2+ bx+c=0 )∨ (ay2+ by+c=0)) ∧ (∀w(aw2+ bw+c=0) $\rightarrow$ (w=x ∨ w=y)]

Use quantifiers and logical connectives to express the fact that a quadratic polynomial with real number coefficients has at most two real roots.answer I think is ∀a∀...

khushtak

1.6k views

khushtak asked Jun 1, 2016

2 votes

2 answers

1090

Kenneth Rosen Edition 6th Exercise 1.3 Question 60,61(Page No. 50)
I can solve these ques. with logical reasoning(apti.) but I need to know how to solve questions 60-d and/or 61-e. 60 a. ∀ x [P(x) $\rightarrow$ Q(x) ] b. ∃ x [R(x) ⋀ $not$ Q(x)] c ∃ x [R(x) ⋀ $not$ ... ) $\rightarrow$ $not$ S(x) ] c ∀ x [ $not$Q(x) $\rightarrow$ S(x) ] d ∀ x [P(x) $\rightarrow$ $not$ R(x) ]

I can solve these ques. with logical reasoning(apti.) but I need to know how to solve questions 60-d and/or 61-e. 60 a. ∀ x [P(x) $\rightarrow$ Q(x) ]b. ∃ x [...

khushtak

439 views

khushtak asked May 27, 2016

3 votes

2 answers

1091

Kenneth Rosen Edition 6th Exercise 1.2 Question 30 (Page No. 29)
Show that (p ∨ q) ∧ (¬p ∨ r) → (q ∨ r) is a tautology. Prove it without Truth tables.

Show that (p ∨ q) ∧ (¬p ∨ r) → (q ∨ r) is a tautology. Prove it without Truth tables.

Shyam Singh 1

931 views

Shyam Singh 1 asked May 23, 2016

2 votes

1 answer

1092

Kenneth Rosen Edition 6th Exercise 1.2 Question 15 (Page No. 23)
Each inhabitant of a remote village always tells the truth or always lies. A villager will give only a Yes or a No response to a question a tourist asks. Suppose you are a tourist visiting this area and come to a ... is standing at the fork in the road. What one question can you ask the villager to determine which branch to take?

Each inhabitant of a remote village always tells the truth or always lies. A villager will give only a “Yes” or a “No” response to a question a tourist asks. Supp...

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