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DISCRETE STRUCTURES:

Answers must be correct. Or else it will be flagged. All of these sub parts need to be answered with step by step process showing all work and reasoning.

DISCRETE STRUCTURES:

1.a. Formalization Formalize the following statements using the connectives A, V, and and the quantifiers V and Every non-negative integer less than 4 is a square. For every real numbers z and y there is a rational umber q such that if r is strictly smaller thany then

Explanation / Answer

1.A. Every non negative integer less than 4 is a square.

Formalization using connectives and quantifiers--

let p= any integer less than 4.

q= p is non-negative integer.

r= The integer is a square.

ANS-- p ^ q -> r

For every real number x and y, there is a rational number q such that if x is strictly smaller than y then x<q<y.

Formalization using connectives and quantifiers--

Let a= x & y are real numbers.

b= x is strictly less than y.

c= q is a rational number less than y but greater than x.

ANS-- a ^ b -> c

There is a smallest non negative integer.

Formalization using connectives and quantifiers--

Let p= It is any integer.

q= It is a negative integer.

z= It is smallest integer.

ANS-- p^(~q) -> ~z

1.B. Let Q(x,y) be the predicate x+y=3.

case 1: For all x belongs to Z

For all y belongs to Z

This quantification denotes the following proposition-

For every pair x,y belonging to Z, Q(x,y) is true.

Clearly, this proposition is FALSE.

case 2: For all x belongs to Z

For some y belongs to Z

This quantification denotes the following proposition-

For every x there is a y belonging to Z such that , Q(x,y) is true.

Given a real number x, there is a real number y such that x+y=3.Hence, this proposition is TRUE.

case 3: For some x belongs to Z

For some y belongs to Z

This quantification denotes the following proposition-

For some x there is some y belonging to Z such that , Q(x,y) is true and x+y=3.Hence, this proposition is TRUE.

NOTE: THE ORDER IN WHICH QUANTIFIERS APPEAR DETERMINES THE VALIDITY OF THE PREDICATE.

1.c. Let the given statement be-

For some x belongs to real numbers,

For all y belongs to real numbers,

(y)^2 >= x.

The negation of the above statement is--

For all x belongs to real numbers,

For some y belongs to real numbers,

(y)^2 < x.

Explanation- For every y in real numbers, there is some x number which is less then the square of y number.This is clearly not possible.Hence the negation of the given statement is FALSE.

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