Write the formal statements for each of the informal statement given below; each

Question

Write the formal statements for each of the informal statement given below; each statement is separate and has no relationships with other statements. Hint:Use predicate logic in order to write the formal statements
0. The only prime number that is even is 2.
Hint: You can use the boolean function prime(n) which returns true if n is a prime number.
1. Let A be an array of integers.
(a) A contains only zeroes and ones.
(b) The element at every third position (positions which are multiples of 3) is a zero.
Hint: Use the notation A(i) to represent the ith value of A and #A to represent the size of the array A. The first element of the array is at position 1, not at zero.
2. Every person likes at least one color.
Hint: Assume that the types Person and Color and the boolean function likes( person,color) are all defined.
3. In a computer system, no two users have the same user names.
Hint: Assume the type User and the function username (u) are already de ned; the function returns the user name of the user u.

Explanation / Answer

statement: The only prime number that is even 2

Even(n): n is even
Prime(n):n is prime

means 2 is an even prime and there exists no other even prime

Predicate logic: (Prime(2) Even(2)) V ¬ ( n ( Prime(n) Even (n) (n 2 ) ) )

2. A is a binary number since only zeros and ones in it

A(i) : i is a number

#A is the size of Array

Given A contains zeros or ones

it can be written as i: 0i<N: A(i)=0,1

next element at every third position (positions which are multiples of 3) is a zero.

i: 3i<3n: A(i)=0

so predicate logic is i(0i<N: A(i)=0,1) V i ( A(i)=0 (3i<3n) )

3

likes(person,color)

the truth of the.above function depends upon the values provided

Given every person likes atleast one color

means there exists a color for all person which they like

(Prime(2) Even(2)) V ¬ ( n ( Prime(n) Even (n) (n 2 ) ) )

can be defined as for every person there exists a color he likes

so predicate logic will be

Person( Color( Likes(Person,Color) ->true) )

4.

means a particular user name can be given to a user only

defined as

there exists a username for every users where u1 not equal to u2

Person( Color( Likes(Person,Color) ->true) )

User( u( username(u) (u1 u2 ) )

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