Determine if each of the following is statement, open statement, or neither. (a)

Question

Determine if each of the following is statement, open statement, or neither. (a) 2^300 > 3^200 (b) x^3 - 3x^2 + 4x - 6. (c) x^3 - 3x^2 + 4x - 6. (c) x^3 - 3x^2 + 4x - 6 = 0. (d) 853 = (56)^15 + 13. (e) There is a prime integer larger than 10^10^26. In each of the following problems, an open statement P(a) is given. Recall that these assertions do not carry a truth value until a value is assigned to the variable a. For each open statement, determine all values of a for which P(a) is true. If no such value exists, state "no such value." (a) 3

Explanation / Answer

2.1.1)

a)This is clearly a “statement” as it will yield a True / False value

b) This is “neither” as the equation doesn’t equate to anything. Which mean it cannot have a truth value

c) This is an open statement as it cannot have a truth value unless a value is assigned to x

d) This is clearly a “statement” as it will yield a True / False value

e) This is clearly a “statement” as it will yield a True / False value

2.1.2)

a) 3<2 & a-6

for this statement to be true, 3<a-6

or a>3+6 = 9 ie., a>9

b) a=4 or 2<3

This statement is true regardless of the value of a, as the other sub statement (2<3) is already true

As such, a R [R is any real number]

c) a=4 or 3<2

For this statement to be true, the sub-statement a = 4 has to be true, thus a = 4

d) If a=4 then 2<3

As the second subs-statement is true, this statement will be true regardless of the truth value of the first one, therefore, a R

e) If a=4 then 3<2

As the second subs-statement is false, this statement will be true only if the truth value of the first one is false. Hence, a = {R – 4} i.e, it can have any real value except 4

f) If 2<3 then a = 6

Here, as the first sub-statement is true, the second one has to be true for this whole statement to be true; as such, a=6

g) If 3<2 then a = 6

As the first subs-statement is false, this statement will be true regardless of the truth value of the second one, therefore, a R

h) a=4 if and only if 2<3

For this statement to be true, the truth value of both the sub-statement must be the same. Now, as the truth value of the second sub-statement is true, the first one must be true as well. Hence a = 4

i)3 <2 if and only if a =6

For this statement to be true, the truth value of both the sub-statement must be the same. Now, as the truth value of the first sub-statement is false, the second one must be false as well. Hence a = {R – 6} i.e, it can have any real value except 6

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