Homework 6

1. Show that there does not exist a real number $t$, such that $t \leq 5$ and $2t \geq 12$.

2. Show that there does not exist a real number $x$, such that for all real numbers $y$, $y \le x$.

3. Let $a$ be a real number. Show that $3a+2<11$ if and only if $a<3$.

4. Let $t$ be an integer. Show that $t ^ 2 + t + 3 \equiv 0 \pmod 5$ if and only if $t$ is congruent to 1 or 3 mod 5.

5. Prove that $P \land Q$ is logically equivalent to $Q \land P$. (Submit only a Lean solution -- no written solution.)

6. Show that $(\exists x, P(x)) \land Q$ is logically equivalent to $\exists x, (P(x) \land Q)$. (Submit only a Lean solution -- no written solution.)