Anna University Coimbatore 2009 B.E Computer Science and Engineering 2 ks with Answer for Discrete Maths(Unit-2) - Question Paper

UNIT II

PREDICATE CALCULUS

PART-A

 

1.     Give an example to show that need not

be a conclusion from .

Solution :

Let . Similarly is defined.

Let A={1} and B={2}.

Since A and B are non-empty and is True.

But is False.

 

2.     Let p(x) denote the statement x>4. What are the truth values of P(5) and P(2)?

Solution :

We obtain the statement P(5) by setting x=5 in the statement x>4. Hence

P(5), which is the statement 5>4 is true. However P(2), which is the statement

2>4 is false.

 

3.     Let Q(x,y) denote the statement x=y+2, what are the truth values of the prepositions Q(1,2) and Q(2,0).

Solution :

To obtain Q(1,2), set x=1 and y=2 in the statement Q(x,y).

Hence Q(1,2) is the statement 1=2+2 which is false.

Similarly for Q(2,0), the statement is 2=0+2 which is True.

 

4.     What are the truth values of the preposition R(1,2,3) and R(0,0,1)?

Solution :

The preposition R(1,2,3) is obtained by setting x=1, y=2 and z=3 in the statement R(x,y,z) which denote x+y=z.

The preposition R(1,2,3) is 1+2=3 which is True.

The preposition R(0,0,1) is 0+0=1 which is False.

 

5.     Find the truth value of where

with universe of discourse E being E={2,3,4}.

Solution :

is True , is False

Therefore is False.

Since R(2), R(3), R(4) are all False, is also False.

Hence is also False.

 

 

6.     Express the statement For every x there exist a y such that in

symbolic form.

The symbolic form is

 

7.     Define Simple statement function. (OR)

Define statement function of one variable. When it will become a statement?

A simple statement function of a variable is defined to be an expression consisting

of a predicate symbol and an individual variable. Such a statement function becomes a

statement when the variable is replaced by the name of any object.

Example : If X is a Teacher is denoted by T(x), it is a statement function. If X

is replaced by John, then John is a teacher.

 

8.     Give the symbolic form of the statement

Every book with a blue cover is a Mathematics book.

The symbolic form is

where S(x) : x is every book with a blue cover

P(x) : Mathematics book.

 

9.     Define Quantifiers.
Certain declarative sentences involve words that indicate quantity such as all, some, none, one. These words help to determine the answer to the question How many? Since such words indicate quantity they are called quantifiers.

 

10.                     Write the following sentence in a symbolic form Every one who is healthy can do all kinds of work.

H(x) : x is a healthy person

H(y) : y is a kind of work

D(x,y) : x can do y

 

11.                     Symbolize the following statement with and without using the set of positive integer on the universe of discourse

Given any positive integers , there is a greater positive integer.

Solution :

Let the variable x and y be restricted to the set of positive integers. Then the

above statement can be paraphrased as follows:

For all x, there exist a y such that y is greater than x. If G(x,y) is x is

greater than y, then the given statement is .

If we do not impose the restriction on the universe of discourse and if we

write P(x) for x is a positive integer, then we can symbolize the given

statement is

 

12.                     Rewrite the following using quantifiers. Some men are genius

M(x) : x is a man

G(x) : x is genius

13.                     Symbolize the expression All the world loves a lover

In first note that the quotation really means that everybody loves a lover. Now

let P(x) : x is a person

L(x) : x is a lover

R(x,y) : x loves y

The required expression is

 

14.                     Identify the bound variables and the free variables in each of the following expressions (a) .

(b) .

Solution :

In (a) the scope of is , while the occurrence of y is a free

occurrence and is free.

In (b) the scope of is , while the occurrence of z is a free

occurrence.

 

15.                     Use quantifiers to express the associate law for multiplication of real numbers.

Solution : where the universe of discourse for x,y,z is

the set of real numbers.

 

16.                     Let the universe of discourse be E={5,6,7}. Let A={5,6} and B={6,7}. Let P(x) : x is in A; Q(x) : x is in B and R(x,y) : x+y<12. Find the truth value of .

Solution : the only possibility is

5 + 6 < 12

is true.

 

 

 

 

 

 

 

17.                     Give an example in which is true but is false.

Solution : Let E = {2,3,5}

Let P(x) : x < 4, Q(x) : x > 6

P(2) is true. is true.

For any x in E, Q(x) is false.

Hence is false.

P(5) is false and Q(5) is false.

is true.

is true.

 

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