Question: Are the following true of false? Answer T or F.

1. \(\exists x\in \mathbb{R}\) such that x ² + x - 30 = 0.
2. \(\forall z\in \mathbb{R},\exists y\in \mathbb{R}\) such that \(z
3. x = 3 is a counter example for " \(\forall \) x \(\in \mathbb{R}\) if x> 5 then x ² > 36.
4. \(\left\{ 1,2 \right\}\in \left\{ 1,2,3 \right\}\)
5. For all sets A, \(\varnothing \in \) A.
6. {(1, 2), (2, 3), (3, 1)} is a one to one function.
7. 6 is in the range of the function \(f:\mathbb{R}\backslash 1\to \mathbb{R}\) defined by \(\frac{2x+1}{x-1}\)
8. For all sets A and B, |A| \(\le \) |A \(\cup \) B|
9. For all sets A and B, if |A| = |B| then |P(A)| = |P(B)|.
10. |[0, 1]| = |(0, 1)| (Both [0, 1] and (0, 1) are intervals.)
11. \(|\mathbb{R}|\le |P\left( \mathbb{Q} \right)|\)
12. \(\mathbb{N}<|\mathbb{N}\times \mathbb{N}|\)
13. The negation of R is anti-symmetric is " \(\exists \) x and y such that x R y and y R x and x \(\ne \) y.
14. The relation < on N is transitive.
15. There is a relation which is neither symmetric nor anti-symmetric.

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