Question: (2 points each) Are the following true of false? Answer T or F.

(a) \(\exists x\in \mathbb{R}\) such that x² + x - 30 = 0.

(b) \(\forall z\in \mathbb{R},\exists y\in \mathbb{R}\) such that \(z<y\) and \(y<{{z}^{2}}\).

(c) x = 3 is a counter example for “ \(\forall \) x \(\in \mathbb{R}\) if x> 5 then x² > 36.

(d) \(\left\{ 1,2 \right\}\in \left\{ 1,2,3 \right\}\)

(e) For all sets A, \(\varnothing \in \) A.

(f) {(1, 2), (2, 3), (3, 1)} is a one to one function

(g) 6 is in the range of the function \(f:\mathbb{R}\backslash 1\to \mathbb{R}\) defined by \(\frac{2x+1}{x-1}\)

(h) For all sets A and B, |A| \(\le \) |A \(\cup \) B|

(i) For all sets A and B, if |A| = |B| then |P(A)| = |P(B)|.

(j) |[0, 1]| = |(0, 1)| (Both [0, 1] and (0, 1) are intervals.)

(k) \(|\mathbb{R}|\le |P\left( \mathbb{Q} \right)|\)

(l) \(\mathbb{N}<|\mathbb{N}\times \mathbb{N}|\)

(m) 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.

(n) The relation < on N is transitive.

(o) There is a relation which is neither symmetric nor anti-symmetric.