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Set Theory. Notation. S={a, b, c} refers to the set whose elements are a, b and c . a S means “a is an element of set S” . d S means “d is not an element of set S” . {x S | P(x) } is the set of all those x from S such that P(x) is true. E.g., T={x  Z | 0<x<10 } .

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notation
Notation
  • S={a, b, c} refers to the set

whose elements are a, b and c.

  • aS means “a is an element of set S”.
  • dS means “d is not an element of set S”.
  • {x S | P(x)} is the set of all those x from S such that P(x) is true. E.g., T={x Z | 0<x<10} .
  • Notes:

1) {a,b,c}, {b,a,c}, {c,b,a,b,b,c} all represent the same set.

2) Sets can themselves be elements of other sets, e.g., S={ {Mary, John}, {Tim, Ann}, …}

relations between sets

B

A

Relations between sets
  • Definition: Suppose A and B are sets. Then

A is called a subset of B: A  B

iff every element of A is also an element of B.

Symbolically,

A B  x, if xA then x B.

  • A  B  x such that xA and xB.

A

A

B

B

A  B

A  B

A  B

relations between sets4
Relations between sets
  • Definition: Suppose A and B are sets. Then

A equals B: A = B

iff every element of A is in B and

every element of B is in A.

Symbolically,

A=B  AB and BA .

  • Example: Let A = {mZ | m=2k+3 for some integer k};

B = the set of all odd integers.

Then A=B.

operations on sets
Operations on Sets

Definition: Let A and B be subsets of a set U.

1. Union of A and B: A  B = {xU | xA or xB}

2. Intersection of A and B:

A  B = {xU | xA and xB}

3. Difference of B minus A: BA = {xU | xB and xA}

4. Complement of A: Ac = {xU | xA}

Ex.: Let U=R, A={x R | 3<x<5}, B ={x R| 4<x<9}. Then

1) A  B = {x R | 3<x<9}.

2) A  B = {x R | 4<x<5}.

3) BA = {x R | 5 ≤x<9}, AB = {x R | 3<x ≤4}.

4) Ac = {xR | x ≤3 or x≥5}, Bc = {xR | x ≤4 or x≥9}

properties of sets
Properties of Sets
  • Theorem 1 (Some subset relations):

1) AB  A

2) A  AB

3) If A  B and B  C, then A  C .

  • To prove that A  B use the “element argument”:

1. suppose that x is a particular but arbitrarily chosen element of A,

2. show that x is an element of B.

proving a set property
Proving a Set Property
  • Theorem 2 (Distributive Law):

For any sets A,B and C:

A  (B C) = (A  B)  (A  C) .

  • Proof: We need to show that

(I)A  (B  C)  (A  B)  (A  C) and

(II) (A  B)  (A  C)  A  (B  C) .

Let’s show (I).

Suppose x  A  (B  C) (1)

We want to show that x  (A  B)  (A  C) (2)

proving a set property8
Proving a Set Property
  • Proof (cont.):

x  A  (B  C)  x  A or x  B  C .

(a) Let x  A. Then

x  AB and x  AC  x  (A  B)  (A  C)

(b) Let x  B  C. Then xB and xC.

Thus, (2) is true, and we have shown (I).

(II) is shown similarly (left as exercise). ■