List of examples and applications

List of examples and applications - Spring 2026

This list contains all examples for concepts and applications of theorems which you can be asked for in Part 1 of the exam. The specific sets and numbers in the questions are just examples; these can vary in the actual questions.

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Logic

Possible examples:

• Consider the following predicates: $\mathrm{M}(x)$: ”$x$ is a mouse” and $\mathrm{C}(x)$: ”$x$ likes cheese”. Express the following statements using a formula:
  • “Every mouse likes cheese.”
  • “Some mice like cheese.” (Please use the correct syntax of formulas.)

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Sets

Possible examples:

• Give an example for a subset of $A=\{1,2,3,4,5\}$.
• Find the union / intersection / difference / symmetric difference / Cartesian product of the sets $A=\{1,2,a\}$ and $B=\{a,b\}$.
• Let $U=\{a,b,c,d,e,f,g,h\}$ be the universal set. Find the complement of $A=\{a,c,e\}$.
• Write down the power set of $A=\{0,2,4\}$.
• How many subsets does the set $A=\{a,b,c,d,e,f\}$ have?

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Relations

Possible examples:

• Give an example for a partition of the set $A=\{1,2,a,b,c,d,e,f,g,h\}$.
• Consider the sets $A=\{1,2,3,4,5\}$ and $B=\{a,d,c,d,e,f,g\}$. Give an example for a binary relation from $A$ to $B$.
• Consider the binary relation $R=\{(0,4),(1,3),(1,4),(3,1),(3,4),(5,2)\}$. Find:
  • the domain of $R$,
  • the range of $R$,
  • the inverse of $R$.
• Consider the binary relation $R=\{(1,2),(1,4),(2,2),(2,4),(3,1),(5,2),(5,5)\}$ and set $A=\{1,4,5\}$. Find:
  • the restriction of $R$ to $A$,
  • the image of $A$ under $R$,
  • the inverse image of $A$ under $R$.
• Consider the binary relation $R=\{(1,3),(2,1),(3,1),(5,3)\}$. Give an example for a binary relation $S$ that is:
  • an extension of $R$,
  • a restriction of $R$.
• Let $A=\{1,2,3,4,5\}$. Give an example for a binary relation on $A$ that is:
  • reflexive,
  • irreflexive,
  • symmetric,
  • antisymmetric,
  • strictly antisymmetric,
  • transitive.
• Give an example for an equivalence relation $\sim$ on a (finite or infinite) set $X$. Please specify both the equivalence relation $\sim$ and the set $X$.
• Give an example for a partial order $⪯$ on a (finite or infinite) set $X$. Please specify both the partial order $⪯$ and the set $X$.
• Consider the sets $A=\{1,2,3,4\}$ and $B=\{a,b,c,d,e,f\}$. Give an example for a function from $A$ to $B$.
• Consider the sets $A=\{1,2,3,4\}$ and $B=\{a,b,c,d\}$. Give an example for:
  • an injective function from $A$ to $B$,
  • a surjective function from $A$ to $B$,
  • a bijective function from $A$ to $B$.
• Consider the binary relations $R=\{(1,2),(1,4),(2,3),(2,4)\}$ and $S=\{(1,2),(1,3),(4,1)\}$. Find $R\circ S$.

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Complex numbers

Possible examples:

• Let $z=-3-5i$ and $w=-1+4i$. Find $z+w$ and $zw$ in algebraic form.
• Let $z=2-5i$. Find $|z|$ and $\overline{z}$.
• Let $z=3(\cos 30^{\circ }+i\sin 30^{\circ })$ and $w=5(\cos 12^{\circ }+i\sin 12^{\circ })$. Find $zw$ and $\frac{z}{w}$ in polar form.
• Let $z=2(\cos 15^{\circ }+i\sin 15^{\circ })$. Write down $z^4$ and the $3^{\text{rd}}$ roots of $z$ in polar form.

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Combinatorics

Possible examples:

• Write down the first four rows of Pascal’s triangle.
• How many:
  • permutations without repetition,
  • 4‑variations without repetition,
  • 4‑variations with repetition,
  • 4‑combinations without repetition,
  • 4‑combinations with repetition does the set $A=\{1,2,\dots ,10\}$ have? (You can leave factorials in your answer, you do not need to work it out as a single number.)
• How many permutations with repetition, with repetition numbers $k_1=2,k_2=1,k_3=4,k_4=4$, do the elements $a,b,c,d$ have? (You can leave factorials in your answer, you do not need to work it out as a single number.)
• Write down the binomial expansion of $(x+y{)}^5$.
• Write down the Inclusion‑exclusion principle for four sets.

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Graphs

Possible examples:

• The degrees of the vertices of a 5‑vertex undirected graph $G$ are the following: $7,7,6,3,3$. How many edges does $G$ have?