(1)Sets Definition A set is an unordered collection of objects called elements or members of the set

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Sets

Definition A set is an unordered collection of objects called elements or members of the set.

Examples:

I S ={1,2,5}

I V ={a,e,i,o,u}

I X ={apple,32,A,Z}

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

I Familiar sets: N,Z,R,Q,R⁺.

I Two common questions

I doesx∈S?

I isX =Y?

Roster method vs set-builder method.

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Sets

Examples:

I S ={1,2,5}

I V ={a,e,i,o,u}

I X ={apple,32,A,Z}

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

I doesx∈S?

I isX =Y?

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Sets

Examples:

I S ={1,2,5}

I V ={a,e,i,o,u}

I X ={apple,32,A,Z}

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

I doesx∈S?

I isX =Y?

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Sets

Examples:

I S ={1,2,5}

I V ={a,e,i,o,u}

I X ={apple,32,A,Z}

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

I Familiar sets:

N,Z,R,Q,R⁺.

I doesx∈S?

I isX =Y?

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Sets

Examples:

I S ={1,2,5}

I V ={a,e,i,o,u}

I X ={apple,32,A,Z}

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

I Familiar sets:

N,Z,R,Q,R⁺.

I doesx∈S?

I isX =Y?

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Sets

Examples:

I S ={1,2,5}

I V ={a,e,i,o,u}

I X ={apple,32,A,Z}

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

I Familiar sets:

N,Z,R,Q,R⁺.

I doesx∈S?

I isX =Y?

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Equality and containment

A=B

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

I Express equality using quantifiers.

∀x (x ∈A↔x ∈B)

A⊆B

I A={a,b,22,33}

I B ={a,b,22,33,N}

I B *A

I Express not a subset of using quantifiers.

∃x (x ∈B∧x∈/ A)

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Equality and containment

A=B

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

∀x (x ∈A↔x ∈B)

A⊆B

I A={a,b,22,33}

I B ={a,b,22,33,N}

I B *A

∃x (x ∈B∧x∈/ A)

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Equality and containment

A=B

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

∀x (x ∈A↔x ∈B)

A⊆B

I A={a,b,22,33}

I B ={a,b,22,33,N}

I B *A

∃x (x ∈B∧x∈/ A)

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Equality and containment

A=B

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

∀x (x ∈A↔x ∈B)

A⊆B

I A={a,b,22,33}

I B ={a,b,22,33,N}

I B *A

∃x (x ∈B∧x∈/ A)

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Equality and containment

A=B

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

∀x (x ∈A↔x ∈B)

A⊆B

I A={a,b,22,33}

I B ={a,b,22,33,N}

I B *A

∃x (x ∈B∧x∈/ A)

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Equality and containment

A=B

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

∀x (x ∈A↔x ∈B)

A⊆B

I A={a,b,22,33}

I B ={a,b,22,33,N}

I B *A

∃x (x ∈B∧x∈/ A)

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Equality and containment

A=B

I Y ={2,3,4,6,22}

I Y⁰ ={2,3,4,3,22,2,6}

I Y⁰⁰={22,4,6,22,3}

∀x (x ∈A↔x ∈B)

A⊆B

I A={a,b,22,33}

I B ={a,b,22,33,N}

I B *A

∃x (x ∈B∧x∈/ A)

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Empty set

Definition A set that has no elements is the empty set denoted by φor{}.

I for any setS,φ∈S.

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Empty set

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Empty set

I for any setS,φ⊆S.

proof?

I ∀x (x ∈φ→x∈S)

I for anyx,x∈φalways evaluates to false.

I ∴the conditional (x∈φ→x ∈S) evaluates to true.

Example of a vacuous proof.

Important {φ}is not the same as φ.

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Empty set

I for any setS,φ⊆S. proof?

I ∴the conditional (x∈φ→x ∈S) evaluates to true.

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Empty set

I ∴the conditional (x ∈φ→x ∈S) evaluates to true.

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Empty set

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Empty set

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Power set

S ={32,6,41}

I subsets ofS:

{},{32},{6},{41},{32,6},{6,41},{32,41},{32,6,41}.

Definition The set of all subsets of S is called the power setof S, denoted asP(S).

The power set of a set withn elements has 2ⁿ elements.

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Power set

S ={32,6,41}

I subsets ofS:

{},{32},{6},{41},{32,6},{6,41},{32,41},{32,6,41}.

The power set of a set withn elements has 2ⁿ elements.

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Power set

S ={32,6,41}

I subsets ofS:

{},{32},{6},{41},{32,6},{6,41},{32,41},{32,6,41}.

The power set of a set withn elements has 2ⁿelements.

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Size of a set

Definition If there are exactlyn distinct elements in a set S, wheren is a non-negative integer, then we say S is finite and cardinality ofS is n.

I S ={1,17,a,b};|S|= 4.

I S ={1,2,2,6,7,7};|S|= 4.

I |φ|= 0.

A set that is not finite is said to be infinite

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Size of a set

I S ={1,17,a,b};|S|= 4.

I S ={1,2,2,6,7,7};|S|= 4.

I |φ|= 0.

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Size of a set

I S ={1,17,a,b};|S|= 4.

I S ={1,2,2,6,7,7};|S|= 4.

I |φ|= 0.

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Set operations

I Union

I Intersection

I Set difference

I Exclusive OR

I Complement Set identities

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Exercises

1. For each of the sets below determine if 2 belongs to the set.

(a){x ∈R | x is an integer greater than 1}

(b) {x ∈R | x is the square of an integer}

(c) {2,{2}} (d){{2},{2}} (e) {{2},{2,{2}}}

2. Let A={φ,b}. Construct the following sets:

I A−φ

I A∪ P(A)

I {φ} −A

I A∩ P(A) 3. Prove thatP(A)⊆ P(B) iff A⊆B.

4. ExpressA⊂B using quantifiers.

5. Prove thatA∩B = ¯A∪B.¯