# 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 $\preceq$ on a (finite or infinite) set $X$. Please specify both the partial order $\preceq$ 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 $A = \{1,2,\ldots,10\}$ 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?