Discrete Math: Set Theory and Functions

Let $R$ be the relation on the set of ordered pairs of positive integers such that $(a, b) \, R \, (c, d)$ if and only if $ad = bc$. Show that $R$ is an equivalence relation.

Given the sets X = {1, 2, 3, 4, 5} and Y = {a, b, c, d, e}, determine whether a mapping f: X \rightarrow Y is surjective or injective. Show examples for each case.

Write an exponential function to model each situation and find the amount after the specified time. For example, given a population of 1,236,000 that grows at an interest rate of 1.3% over 10 years, find the final population.

Prove the function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = \pi x - e$ is a bijection and find the inverse.

Prove that a function $f : [0, 1] \to [2, 4]$ is a bijection by showing it is well-defined, injective, and surjective.

We have 150 students who drink three beer brands: A, B, and C. Given: 58 students drink brand A, 49 drink brand B, 57 drink brand C, 14 drink both A and C, 13 drink both A and B, 17 drink both B and C, and 4 drink all three brands A, B, and C. Determine how many students drink none of these brands.

Show that the complement of $A_1 \cup A_2 \cup \ldots \cup A_n$ is equal to the complement of $A_1$ intersect the complement of $A_2$ intersect all the way up to the complement of $A_n$.

Find the power set of the set $A = \{ 1, 2, 3 \}$.

Find two sets $A$ and $B$ such that $A$ is an element of $B$ and $A$ is a subset of $B$.

Determine if the following sets are power sets of some unknown set. For example, for a given set, check if it can be the power set by checking the number of elements which should be a power of two.

Prove that $(A - B) \cap (B - A) = \emptyset$.

Prove that the intersection of $A$ and $B$ is a subset of $A$ union $B$.

Prove that $A - B \cap C$ is a subset of $A - B \cup A - C$.

Prove De Morgan's Law: $(A \cup B)' = A' \cap B'$.

Prove that if $A$ is a subset of $B$, then $A - C$ is a subset of $B - C$.

Write an equivalent logical expression using quantifiers for the statement: "A union B is a subset of C difference D".

Write an equivalent logical expression using quantifiers for the statement: "A union B is not a subset of C difference D" using the negation of previous statements.

Given a set with elements $A, B, C, D$, determine if the relation is reflexive, symmetric, and transitive by considering the arrows between the elements.

Verify if the given relation is transitive by checking all possible paths between elements.

Given a set $A$ with elements, determine if the relation on $A$ is reflexive, symmetric, or transitive based on the arrow diagram provided.