4. For the following statement, first write the formal statement and then its ne

Question

4. For the following statement, first write the formal statement and then its negation [10 points]

For all integers n, if n is prime then n is odd

Formal statement:

Negation:

5. For the following statement (1) write the statement informally without using variables or the symbols

V(for all) or 3(there exists), and (2) indicate whether the statement is true or false and briefly justify your answer.

v(for all) integers a, 3(there exists) an integer b such that a + b = 0.

6. Is the following argument valid or invalid? Justify your answer.

All real numbers have nonnegative squares,

The number i has a negative square

Therefore, the number i is not a real number.

7. Define the predicate F(x; y): x is a factor of y. Find the truth set of F(x, y) when the domain of x and y is

{13, 6, 8, 16}. Note that your truth set should consist of ordered pairs (x, y) that satisfy the given predicate.

8. Use a diagram to show that the following argument can have true premises and a false conclusion.

All dogs are carnivorous.

Aaron is not a dog.

Aaron is not carnivorous.

Explanation / Answer

(4) For all integers n, if n is prime then n is odd.

  Formal statement : n, (P -> Q)

where P is ' n is prime ' and Q is ' n is odd '.

represents 'for all'.

  negation :   n, (P -> Q) = P Q.

(5) For all integers n, if n is prime then n is odd in sentence form.

for all n, n is prime -> n is odd

negation : for all n,   (n is prime -> n is odd) = n is prime n is odd.

(6)

All real numbers have nonnegative squares, -- 1

The number i has a negative square --2

Therefore, the number i is not a real number. --3

-> This is invalid argument.Because, 1 is correct,2 is false because if we square any number then it becomes positive. i is a negative number and real number,because real numbers can accept both negatives and rations.

so,3 is also false.so the given arguments are invalid.

(7) predicate F(x; y): x is a factor of y. Find the truth set of F(x, y) when the domain of x and y is

{13, 6, 8, 16}.

factors of 13 : 1 and 13

factors of 6 : 1,2,3 and 6

factors of 8 : 1,2,4 and 8

factors of 16 : 1,2,4,8 and 16

X contains : 13,6,8 and 16,

Y contains : 1,2,3,4,6,8,13,16.

The truth set is F(X,Y) = { (1,13),(13,13),(1,6),(2,6),(3,6),(6,6),(1,8),(2,8),(4,8),(8,8),(1,16),(2,16),(4,16),(8,16), (16,16) } .

/// *** Thank You *** ///

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