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## MTH 4000: Practice Midterm 1

Problem 1. Prove that $\neg \left( \left(P\wedge Q\right)\to \left(R\vee S\right)\right) \equiv \left(P\wedge Q\wedge \neg R\wedge \neg S\right)$

Problem 2. Let $$P$$ and $$Q$$ be two propositions. The proposition $\left(\left(P\leftrightarrow Q\right)\to\left(\neg P\right)\right)\wedge \neg Q$ is equivalent to:
• (A) $$P$$
• (B) $$Q$$
• (C) $$\neg P$$
• (D) $$\neg Q$$
• (E) $$P\wedge Q$$
• (F) $$P\wedge \neg Q$$
• (G) $$\neg P\wedge Q$$
• (H) $$\neg P\wedge \neg Q$$

Problem 3. Determine whether the following proposition is true or false: For every prime number $$p$$, the number $$p^2+4$$ is prime.

Problem 4. Determine which of the following propositions are true and which are false. Provide a rigorous justification for your answers.
• (a) $$\left(\forall x \in \mathbb R_+\right)\left(\exists y\in\mathbb R_+\right)\left(\forall z\in\mathbb R\right)\left( \left(z > y\right) \rightarrow \left(z\cdot x^2 > 1\right)\right)$$.
• (b) $$\left(\forall x \in \mathbb R_+\right)\left(\exists y\in\mathbb R_+\right)\left(\forall z\in\mathbb R\right)\left( \left(z > y\right) \rightarrow \left( x^2 > z\right)\right)$$.
• (c) $$\left(\forall x\in\mathbb R\right)\left(\exists y\in\mathbb R_+\right)\left(\forall z\in\mathbb R\right)\left( \left(z > y\right) \to \left(z^2 \cdot x^2 > y\right)\right)$$.

Problem 5. Find all sets of four consecutive positive integers such that the cube of the largest is the sum of the cubes of the other three.

Problem 6. Prove that a perfect square of an integer is congruent to either $$0$$ or $$1$$ modulo $$3$$, and a perfect square of an integer is congruent to either $$0$$ or $$1$$ modulo $$4$$.

Problem 7. Let $$n$$ be a positive integer. Prove that if $$a$$ and $$b$$ are integers such that $$a\equiv b\; (\mbox{mod }n)$$, then $$a^x\equiv b^x\; (\mbox{mod }n)$$ for every positive integer $$x$$.

Problem 8. Find all pairs of integers $$x$$ and $$y$$ such that $$x+y+xy=80$$.

Problem 9. Determine whether the following proposition is true or false. $\left(\forall k\in\mathbb N\right)\left(\exists m\in\mathbb N\right)\left(\forall n\in\mathbb N\right) \left( (n\geq m) \;\rightarrow \; (n+1)\nmid n^2+k\right).$ Provide a rigorous justification for your answer.

Problem 10. Find all pairs $$(x,y)$$ of positive integers for which $$2014^x+11^x=y^2$$.