Chapter - 2, Quantificational Logic

Section - 2.1 - Quantifiers


  • is used for statements which are true for all possible input values.
  • is used for statements which are true for atleast one input value.
  • Quantifiers binds a variable. A variable that is bound by a quantifier can always be replaced with a new variable without changing the meaning of the statement, and it is often possible to paraphrase the statement without mentioning the bound variable at all.
  • Changing the order of quantifiers can change the meaning of the sentence. is not same as .
  • However, Changing the order of quantifiers will not change the meaning of the sentence if all the quantifiers are same. is not same as .

Solutions

Soln1

(a)
where means x has forgiven y.
and means x is a Saint.

(b)
where means x is student of calculus class.
and means y is student of descrete class.
and means x is smarter than y.

(c)
where means x likes Mary.

(d)
where means Jane saw y.
where means Roger saw y.
where means y is a Police officer.

(e)
where means Jane saw y.
where means Roger saw y.
where means y is a Police officer.


Soln2

(a)
where means Bought Rolls Royce with cash.
and means x is Rich.
and means y is uncle of x.

(b)
where means x lives in Dome,
and means x has Measles,
and y is a friend of x,
and y is quarantined.

(c)
where means x is failed,
and means x got A,
and means x got D,
and means x teaches y.

(d)
where x can do it.

(e)
where means x can do it.


Soln3

(a) . Here is a free variable as it is not bound to any quantifiers.

(b) . No free variables.

(c) . No free variables.

(d) . w is a free variable.


Soln4

(a) All unmarried man are unhappy.

(b) y is sister of x’s parent.


Soln5

(a) All primes numbers except 2 are odd.

(b) There exists a perfect number which is greater than or equal to all other perfect numbers.


Soln6

(a) There is at-least one person who is parent of everyone. False.

(b) Everyone is a parent of atleast one person. False.

(c) There are no parents. False.

(d) There at-least exists one person who is not a parent of anyone. True.

(e) There at-least exists some person who is not a parent of someone. True.


Soln7

(a) For all there exists a such that . As . For every natural number , We have a which is also natural number. True

(b) There exists a such that for every value of , . False.

(c) For all there exists a such that . As . For every natural number , We dont have y as a natural number. eg: x = 3, y is not natural number. False.

(d) It is clear from the statement that if x < 10 then y must be < 9. True.

(e) There exists y and z such that sum of y and z is 100. True.

(f) For x = 200. y > 200. Thus clearly z must be negative for statement to become true. But x,y,z are natural numbers. Thus False.


Soln8

(a) True

(b) False

(c) True. y can take fractional values.

(d) False. Here y can take values like 91/100(9.1), 92/100(9.2).. Thus y < 9 is not true.

(e) True

(f) True. Compared to last question, Now z can be negative. Thus statement is true.


Soln9

(a) True

(b) False

(c) False

(d) True

(e) True

(f) True