1 Section 1.7 Set Operations

2 Union The union of 2 sets A and B is the set containing elements found either in A, or in B, or in both The denotation for A union B is A  B Union is related to disjunction, as follows: A  B = { x | (x  A)  (x  B) }

3 Venn Diagrams A Venn diagram is a pictorial representation of sets and set operations. The background of the diagram represents the universal set, and each set involved in the operation is depicted as a circle The shaded region of the diagram represents the operation shown

4 Intersection The intersection of 2 sets A & B is the set containing only those elements found in both A and B Intersection is denoted with the symbol  Intersection is related to conjunction as union is related to disjunction: A  B = { x | (x  A)  (x  B) } 2 sets are disjoint if their intersection is the empty set

5 Intersection The Venn diagram for intersection:

6 Cardinality of the Union of 2 Sets |A| + |B| is the sum of the number of elements in both sets Since the sets may intersect, we need to subtract the cardinality of the intersection in order to count the elements in each set only once; Thus: |A  B| = |A| + |B| - |A  B| Generalization of this result to an arbitrary number of sets is called the principle of inclusion/exclusion

7 Difference of 2 Sets Denoted A - B, is the set containing only those elements that are in the first set but not in the second set Therefore: A - B = { x | (x  A)  (x  B) }

8 Difference of 2 Sets

9 Let A = {a, b, c, d} and B = {c, d, e, f} Then A - B = {a, b} and B - A = {e, f }

10 Complement of a Set The complement of a set is the set of all elements found in the universal set EXCEPT the elements of the set in question The complement of set A is denoted as follows: A = { x | x  A }

11 Complement of a Set

12 Set Identities The next several slides introduce set identities As will be evident from their names, these identities are analogous to the the logical equivalences we saw earlier

13 Identity Laws A   = A A  U = A

14 Domination Laws A  U = U A   = 

15 Idempotent Laws A  A = A A  A = A

16 Complementation (A) = A

17 Commutative Laws A  B = B  A A  B = B  A

18 Associative Laws A  (B  C) = (A  B)  C A  (B  C) = (A  B)  C

19 Distributive Laws A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C)

20 DeMorgan’s Laws A  B = A  B A  B = A  B

21 Proving Set Identities Method 1: Show that each set is a subset of the other Method 2: Use set builder notation and the rules of logic Method 3: Use membership tables (analogous to truth tables)

22 Proving Set Identities: Subsets The text provides a proof of DeMorgan’s second law; here is a proof of DeMorgan’s first law: A  B = A  B Suppose x  A  B. In other words, x  A  B - therefore, x  A and x  B, and x  A  B. Thus, A  B  A  B Now, suppose x  A  B. This means x  A and x  B, so x  A  B, or x  A  B. Thus, A  B  A  B Since each set is a subset of the other, the two sets must be equal and the identity is proved.

23 Using Set Builder Notation & Rules of Logic Again, a proof of DeMorgan’s first law: A  B = A  B A  B = { x | x  A  B } = { x |  ( x  A  B } = { x |  ( x  A  x  B) } = { x | x  A  x  B } by DeMorgan’s Law = { x | x  A  x  B) } = { x | x  A  B }

24 Using Membership Tables Consider each combination of sets an element can belong to A 1 indicates an element belongs to a set; a 0 indicates the element doesn’t belong Verify that elements in the same combination of sets belong to both sets in the identity

25 One more time, DeMorgan’s Law A  B = A  B ABAB A  BA  BA  B

26 Generalized Unions & Intersections The union of a collection of sets is the set containing those elements that are members of at least one set in the collection The intersection of a collection of sets is the set that contains those elements that are members of all sets in the collection

27 Example LetA = {red, green, blue} B = {red, orange, yellow} C = {red, blue, yellow} A  B  C = {red, green, blue, orange, yellow} A  B  C = {red}

28 Generalized Unions & Intersections The notation for A 1  A 2  …  A n is: n i=1  A i The notation for A 1  A 2  …  A n is: n i=1  A i

29 Computer Representation of Sets Two conditions necessary for set representation using bit strings –U must be finite –we impose an arbitrary ordering on U We use 1 to mean the element is a member of the set, and 0 to mean it is not

30 Example SupposeU = {a, b, c, …, z} and S = {a, b, c, x, y, z} We can represent S as: 1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1 The complement of S is the bit string with the bits reversed: 0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0 To find the union of 2 sets, check for 1 bits in the same position in either of the 2 strings; to find the intersection, a 1 bit in the same position of both strings represents an intersection

31 Section 1.7 Set Operations - ends -