Discrete Math: "For all natural n, n^2 > n": is this true, false, a predicate?

[Plonker]

Discrete Math: "For all natural n, n^2 > n": is this true, false, a predicate?

https://i.gyazo.com/0dba70db4045aeae0a7b654b41405ef5.png

Question 3: For each of the following sentences, say whether the sentence is a true statement, a false statement, or a predicate. Also give the negation of each sentence.

\(\displaystyle \mbox{(a) }\, \forall n\, \in\, \mathbb{N},\, n^2\, >\, n\)

\(\displaystyle \mbox{(b) If }\, x\, <\, 0,\, \mbox{ then there is no }\, y\, \mbox{ such that }\, y^2\, =\, x\)

\(\displaystyle \mbox{(c) }\, \forall p\, \in\, \mathbb{P},\, [\exists n\, \in\, \mathbb{N}\, p\, =\, 4n\, +\, 1\, \Rightarrow\, \exists a,\, b\, \in\, \mathbb{N}\, p\, =\, a^2\, +\, b^2]\)

. . .\(\displaystyle [\mathbb{P}\, \mbox{ denotes the set of all prime numbers.}]\)

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I'm unsure about these three, here are my attempts. Please also explain the difference between a predicate and true/false. I assumed it is a predicate when it can be either true or false.

a) Predicate. Negation is ¬(∃n ∈ N n²>n)
b) True. Negation is, "When x<0 there is y such that y^2=x
c) No clue

Your help is truly appreciated!

 

[HallsofIvy]

https://i.gyazo.com/0dba70db4045aeae0a7b654b41405ef5.png

Question 3: For each of the following sentences, say whether the sentence is a true statement, a false statement, or a predicate. Also give the negation of each sentence.

\(\displaystyle \mbox{(a) }\, \forall n\, \in\, \mathbb{N},\, n^2\, >\, n\)

\(\displaystyle \mbox{(b) If }\, x\, <\, 0,\, \mbox{ then there is no }\, y\, \mbox{ such that }\, y^2\, =\, x\)

\(\displaystyle \mbox{(c) }\, \forall p\, \in\, \mathbb{P},\, [\exists n\, \in\, \mathbb{N}\, p\, =\, 4n\, +\, 1\, \Rightarrow\, \exists a,\, b\, \in\, \mathbb{N}\, p\, =\, a^2\, +\, b^2]\)

. . .\(\displaystyle [\mathbb{P}\, \mbox{ denotes the set of all prime numbers.}]\)

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I'm unsure about these three, here are my attempts. Please also explain the difference between a predicate and true/false. I assumed it is a predicate when it can be either true or false.

a) Predicate. Negation is ¬(∃n ∈ N n²>n)

A "predicate" (also called an "open sentence") is a statement whose truth or falsity depends upon the value of a open variable, with no quantifier, in the statement. Here the statement begins with the quantifier "for all n" so there is no "open" variable. This statement is false because, since it is not true for n= 1, it is not true "for all n". Also the negation you give is incorrect because of that "∃". You can get the negation of a statement by putting "¬" at the beginning of the original statement. That is, \(\displaystyle \not(\forall n\in N, n^2> n)\) though that is probably not what is intended. That inverse is the same as \(\displaystyle \exists n\in N, n^2\le n)\)

b) True. Negation is, "When x<0 there is y such that y^2=x".

This is true provided y is required to be a real number.

c) No clue

Primes of the form 4n+ 1 are 5, 13, 17, 29, etc.. 4+ 1= 5, 9+ 4= 13, 25+ 4= 29. is there any prime number, one more than a multiple of four, that is not the um of two squares?

Your help is truly appreciated!