The following are some notes taken during a lecture.

Set Union

Definition: Let A and B be sets. The union of the sets A and B, denoted by A ∪ B,  is the set:

Set Intersection

Definition:  The intersection of sets A and B, denoted by  A ∩ B,  is BELOW. Note if the intersection is empty, then A and B are said to be disjoint.

Set Complement

Definition: If A is a set, then the complement of the A (with respect to U), denoted by Ā is the set  U – A

Set Difference

Definition: Let  A and B be sets. The difference of A and B, denoted by A – B, is the set containing the elements of A that are not in B. The difference of A and B is also called the complement of B with respect to A.

Cardinality of union of Sets

Examples:

Example: U = {0,1,2,3,4,5,6,7,8,9,10}  A = {1,2,3,4,5},    B ={4,5,6,7,8}

 

• A ∪ B             

 

 Solution: {1,2,3,4,5,6,7,8}     

 

• A ∩ B            

 

 Solution: {4,5} 

 

• Ā                  

 

  Solution: {0,6,7,8,9,10}

1.                         

 Solution: {0,1,2,3,9,10}

 

• A – B            

 

  Solution: {1,2,3} 

 

• B – A               

 

Solution: {6,7,8}

 

Symmetric Difference

Definition: The symmetric difference of A and B, denoted by             is the set

Set Identities

Proving Set Identities

Different ways to prove set identities:

1. Prove that each set (side of the identity) is a subset of the other.
2. Use set builder notation and propositional logic.
3. Membership Tables: Verify that elements in the same combination of sets always either belong or do not belong to the same side of the identity.  Use 1 to indicate it is in the set and a 0 to indicate that it is not.

Proof of DeMorgan’s Laws on Sets

Set Builder Notation Proof of DeMorgan’s Law

 

Proof Using Membership tables

 

Generalized Unions

 

 

 

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