## 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