Clep: Sets

Equivalent Sets

If subsets have the same number of elements. Ex. A= {1,2,3,4,5} B= {1,2,6,4,8} Both have five elements

Equal Sets

If two sets have exactly the same elements( in any order). If A= B, then A ⊆B and vice versa. Ex. A = {1,2,3,4,5} C= { 1,2,3,4,5}

Equivalent sets

If two sets have the same number of elements. Ex. O ={3,7,9,12} E= {4,7,12,19} Both gave four elements.

Example

M = { 1,3,5} N= { 2,8} {(1,2), (1,8), (3,2),(3,8),(5,2),(5,8)}

Sets

Sets: capital letters (A) Elements: lower case letters (k). k∈A means that k is apart of the A set. k ∉A means that k does not belong to set A.

Subset

Subsets have the same elements. A ⊆ B Ex. A={1,2,3,4} B={1,2,3,4}

Difference

The difference of two sets, A and B, written as A-B,is the set of all elements that belong to A but do not belong to B.

Venn Diagram :Union of sets

The union of two sets A and B, denoted A∪B, is the set of all elements that are either in A or B or both.

Venn Diagram: Intersection of sets.

Two sets, A and B, denoted A ∩ B, is the set of all elements that belong to A and B.

FYI

Whenever set A is a proper subset of B, A-B= Φ. If set P is any set, then P -Φ= P and Φ-P=Φ If P and Q are sets, P-Q= P ∩Q'

Sets

A = {whole numbers} {1,2,3,4,5,6,7,8,9}

Disjoint

A and B are disjoint if A∩B= ∅ Contains no common elements. {0,4,8,12} {6,10,14, 18}

Proper Subset

A is a proper subset to B if B contains at least one element not in A. A ⊂ B

Sets: Null or empty set.

A set not containing any elements. It is represented by a ∅ or {}.

Intersection of two sets

A ∩ B * think of a street intersection with a u turn.

Union of two sets

A ∪ B * think "U" for union

Drill Question 1. Suppose that set K has 12 elements and set L has 3 elements. How many elements are there in the Cartesian Product?

A. 4 B. 9 C. 15 D. 36 Answer: D because twelve times three equals thirty six.

Venn Diagram: Distributive Laws

AU(B∩C) = (AUB)∩ (AUC). A∩(BUC) = (A∩B) U (A∩C).

DeMorgan's Laws

(AUB)' = A'∩ B' (AUB)' = A' U B'

Example

If A= {1,2,3,4,5} and B={2,3,4,5,6}, then A ∪B ={1,2,3,4,5,6} and A∩B = {2,3,4,5}.

Venn Diagram: Associative Laws

(A U B) U C= A U (BUC) (A∩B)∩C= A∩ (B∩C).

Examples

1.J= {10,,12,14,16} K={9,10,11,12,13} J-K={14,16} Note: K-J= {9,11,13} 2. T={a,b,c} V= {a,b,c,d,e,f} T-V= Φ, but, V-T= {d,e,f} T is the proper subset of V.

Drill Question 2. If set M has ten elements and Set N has seven elements, what is true about them?

A. The max number of elements in M' is 3 B. The minimum number of elements in M ∩N is 7. C. The max number of elements in M U N is 17. D. The minimum number of elements in M-N is 10. Answer: C because ten plus seven is seventeen.

Infinite Sets

Any set that is not finite. Has 100 elements. {1,2,3,4,....300} Example: {x| x where all real numbers are between 4 and 5}

Venn Diagram: Laws of Set Operations: Identity Laws

A∪ ∅= A A ∪U= U A∩ U =A

Venn Diagram: Complement Laws

A∪A' = U A∩ A' = Φ Φ' = U U'= Φ

Venn Diagram :Laws of set operations: Idempotent Laws

A∪A=A A∩ A=A

Venn Diagram :Commutative Laws

A∪B = B∪A A∩B = B ∩A

Sets Example

B ⊆ A, B ⊆C, B ⊂C, B ⊂ A. B is the subset and proper subset of A and C.

Drill Question If the universal set, U= {x| x is a positive odd integer less than 30}, R= {1,5,7} and S= {1,3,,7,11,13}., how many elements are in (R ∩ S)'? A. 15 B. 13 C. 7 D. 2

B. common elements between R and S= {1,7}, therefore, 2 numbers less than 30= 15 hence 30/2= 15 15-2= 13

If A={ x| x is an even integer less than 103} and B= { all negative numbers]. Which of the following describes A ∩ B? A. { All negative numbers and all positive even numbers} B. { All negative numbers} C. { All negative even numbers} D. { all negative odd numbers}.

C. In roster form, A= {...,-8,-6,-4,-2,0,2,4,6,8}, we can't write B in roster form, but, we can determine the elements common to both A and B {....-8,-6,-4,-2}, which is the set of all negative even integers.

Finite Sets

Counted elements. {1,2,3,4,5}. Has 4 elements.

Drill Question 3. Which of the following are disjoint sets? A. {0,1,2,3,} and { 3,2,1,0} B. { 0,2,4,6} and {2,4,6,8} C. {0,3,6,9} and {9,16,25,36} D. { 0,4,8,12} and { 6,10, 14, 18}.

D.

Universal Set

Ex. U={1,2,3,4,5,6....} A= {1,2,3}, then A' = {4,5,6...}.

Cartesian Product

Given two sets, M and N, denoted M x N, is the set of all ordered pairs of elements in which the first component is a member of M and the second component is a member of N.