21.1.1. Lecture -20

Try these first

1. 1.

   Prove that if A ∩ C ⊆ B ∩ C, then A ⊆ B.

   Hint

   Use the definition of subset and intersect sets.

2. 2.

   Define a symmetric relation.

   Hint

   Think about how pairs can interrelate in a set.

3. 3.

   What does it mean for A ∩ C ⊆ B ∩ C?

   • A ⊆ B
   • A ∩ B ⊆ C
   • C is empty

   Hint

   Think of the implications of the filter analogy we discussed.

4. 4.

   True or False: In a symmetric relation, missing (a,b) implies that (b,a) is also not present.

   • True
   • False

   Hint

   Revisit the definition of symmetry.

5. 5.

   Can there exist a relation on a set with no overlap, yet be both symmetric and irreflexive? Justify your answer.

   Hint

   Contrast the definitions to guide your reasoning.

6. 6.

   Create a relation matrix for 3 elements that exemplifies a reflexive yet symmetric relation.

   Hint

   Remember to ensure all ordered pairs considered satisfy the properties discussed.