# Discrete math

**Question description**

1. Consider: ∀ x ∈ N , ∃ y ∈ N *, *x > y.

a) Translate the statement into an English sentence that contains no mathematical notation.

b) Find the negation of the statement in symbols.

c) State that the original sentence is true or false and prove your assertion by either (pretty) proving the original statement is true or by (pretty) proving the negation of the original statement is true.

2. Let A, B, and C be sets. Prove A x (B∪C) = (A x B ) ∪ ( A x C) using a set containment both directions argument.

3. Consider the relation *R *on the set S = 2^{z},
the power set of Z , defined by A*R*B provided A ∩ B ≠ ∅. Which of the five properties: reflexive, irreflexive, symmetric,
antisymmetric, transitive, does *R *have? Give a “pretty proof” for each
of your answers.

Hints:

1. Think carefully about N being here as opposed to Z .

2. Think about what a typical element of A × (B∪C) “looks like”. Be sure to set up and write a set containment both directions argument the framework for which can be found as ex. 3 on the Day 18 notes. BUT a typical element of A × (B∪C) should not be called x.

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