1. CSE 20: Discrete Mathematics for Computer Science Prof. Shachar Lovett

2. Review problems
Midterm 2

3. Material Everything we studied (excluding 1st midterm)
Graphs, sets, functions, relations
Proofs: direct, contrapositive, by contradiction, induction, strong induction

4. Question 1 Let f:NN be defined as f(x)=(x+1)2. Is f: Injective
Surjective
Bijective
None

5. Question 2
Write the set “all elements which are equal to 1 modulo 3” in set builder notation
{𝑥∈𝑍:∃𝑦∈𝑍, 𝑥=3𝑦+1}
𝑥∈𝑍:∃𝑦∈𝑍, 𝑦=3𝑥+1
{𝑥∈𝑍:∀𝑦∈𝑍, 𝑥=3𝑦+1}
{𝑥∈𝑍:∀𝑦∈𝑍,𝑦=3𝑥+1}

6. Question 3
Let X,Y be finite sets. f:XY be a function such that f is injective but not surjective. Prove that |X|<|Y|.
Proof (direct proof):
Let A={f(x): x X}. Note that AY.
Since f is injective, all the elements f(x) are unique, hence |A|=|X|.
Since f is not surjective, there are elements in Y not attainable as images of f. Hence AY.
As AY and AY, A is a strict subset of Y. Hence |A|<|Y|.
Thus, |X|<|Y|.
QED.

7. Question 4a
Let X=“all formulas  on n inputs x1,…,xn, using the connectives AND,OR,NOT”.
Define a relation R for ,X as “ is a tautology”.
Is R symmetric:
Yes No

8. Question 4b
Let X=“all formulas  on n inputs x1,…,xn, using the connectives AND,OR,NOT”.
Define a relation R for ,X as “ is a tautology”.
Is R reflexive:
Yes No

9. Question 4c
Let X=“all formulas  on n inputs x1,…,xn, using the connectives AND,OR,NOT”.
Define a relation R for ,X as “ is a tautology”.
Is R transitive:
Yes No