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12. Let \( R_{1}, R_{2} \subseteq S \times S \) be two relations. Recall that \( R_{1} \circ R_{2}:=\left\{(a, c) \in S \times S \mid \exists b \in S\right. \) such that \( (a, b) \in R_{1} \) and \( \left.(b, c) \in R_{2}\right\} \). Which of the following are true for all relations? If \( R_{1} \) and \( R_{2} \) are transitive then so is \( R_{1} \circ R_{2} \). If \( R_{1} \) and \( R_{2} \) are symmetric then so is \( R_{1} \circ R_{2} \). If \( R_{2} \) is reflexive then \( R_{1} \subseteq R_{1} \circ R_{2} \). If \( R_{1} \subseteq R_{1} \circ R_{2} \) then \( R_{2} \) is reflexive. 13. Let \( A, B, C \) be finite sets and \( P \subseteq A \times B \) and \( Q \subseteq B \times C \) be relations. Which of the following statements are true? \( |P \circ Q| \leq|P| \cdot|Q| \) \( |P \circ Q| \leq|A| \cdot|Q| \). \( |P \circ Q| \leq|P| \cdot|C| \). \( |P \circ Q| \leq|A| \cdot|C| \).

          12. Let \( R_{1}, R_{2} \subseteq S \times S \) be two relations. Recall that
\( R_{1} \circ R_{2}:=\left\{(a, c) \in S \times S \mid \exists b \in S\right. \) such that \( (a, b) \in R_{1} \) and \( \left.(b, c) \in R_{2}\right\} \). Which of the following are true for all relations?
If \( R_{1} \) and \( R_{2} \) are transitive then so is \( R_{1} \circ R_{2} \).
If \( R_{1} \) and \( R_{2} \) are symmetric then so is \( R_{1} \circ R_{2} \).
If \( R_{2} \) is reflexive then \( R_{1} \subseteq R_{1} \circ R_{2} \).
If \( R_{1} \subseteq R_{1} \circ R_{2} \) then \( R_{2} \) is reflexive.
13. Let \( A, B, C \) be finite sets and \( P \subseteq A \times B \) and \( Q \subseteq B \times C \) be relations. Which of the following statements are true?
\( |P \circ Q| \leq|P| \cdot|Q| \)
\( |P \circ Q| \leq|A| \cdot|Q| \).
\( |P \circ Q| \leq|P| \cdot|C| \).
\( |P \circ Q| \leq|A| \cdot|C| \).

12. Let R1, R2⊆ S × S be two relations. Recall that
R1∘ R2:={(a, c) ∈ S × S |∃ b ∈ S. such that (a, b) ∈ R1 and .(b, c) ∈ R2}. Which of the following are true for all relations?
If R1 and R2 are transitive then so is R1∘ R2.
If R1 and R2 are symmetric then so is R1∘ R2.
If R2 is reflexive then R1⊆ R1∘ R2.
If R1⊆ R1∘ R2 then R2 is reflexive.
13. Let A, B, C be finite sets and P ⊆ A × B and Q ⊆ B × C be relations. Which of the following statements are true?
|P ∘ Q| ≤|P| ·|Q|
|P ∘ Q| ≤|A| ·|Q|.
|P ∘ Q| ≤|P| ·|C|.
|P ∘ Q| ≤|A| ·|C|.

Added by Ismael W.

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