It appears you've uploaded a document with various midterm questions covering topics in set theory, logic, functions, relations, mathematical induction, and more. I can help with any of the specific questions or provide detailed explanations on topics like Venn diagrams, logical connectives, equivalence relations, or anything else listed in your document.

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Here are five questions from your document to get started:

1. How do you prove that $1 + 2 + 2^2 + \dots + 2^n = 2^{n+1} - 1$ using mathematical induction?
2. How can we find the matrix of the relation and inverse relation given in problem 1?
3. What are the differences between reflexive, symmetric, and transitive relations with examples?
4. How do you construct the converse, inverse, and contrapositive of "If you work hard, you will be rewarded"?
5. How can you verify if a function is injective or surjective?

Tip: Always begin by identifying the structure of a problem (whether algebraic, logical, or set-based) before trying to solve it to understand the tools and methods that are most effective.