First Day Definitions

Logic Symbols

Set Builder Notation

Set Logic

100

What is the definition of a positive integer?

An integer x > 0.

100

What is the truth table for

P ↔ Q ?

T

F

F

T

100

Write using set builder notation, the set of positive integers

{xεZ : x>0 }

100

What does x ε (A U B) mean?

x ε A or xεB

200

What is the definition of an **Even Number**?

An integer x, s.t. 2|x is true.

200

Is P → Q ⇔ (¬Q) → (¬P) true?

Yes! This statement even has a fancy name, Proof by Contrapositive!

200

Write using set builder the set of odd negative integers

{ xεZ : x<0 ^ x is odd}

200

What is the difference of two sets?

A-B

For xεA-B,

x is in A and not in B.

300

What does it mean if a divides b? Provide definition and the mathematical symbol that defines the statement.

a|b

a divides b means there exists a third integer c, s.t. b = a*c

300

When is an If-Then statement false?

P → Q

An If-Then statement is only false when the initial condition is true and the second condition is false.

300

Write the set of prime numbers using set builder

{x: x is prime}

300

A ∪ (B ∩ C)= ?

(A ∪ B) ∩ (A ∪ C)

400

What does it mean if p is prime?

If p is prime, p>1 and 1 and p are its only divisors.

400

List a distributive property of logic symbols

P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R)

or

P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R)

400

What is the symbol for the empty set?

Either {} or

400

What is the symmetric difference of two sets?

AΔB

For xεAΔB,

x is in only A or x is in only B.

500

What is a Lemma?

A lemma is a theorem that is used primarily to help solve another theorem.

500

List one of DeMorgan's Laws using logic symbols?

¬(P ∨ Q) ⇔ (¬P) ∧ (¬Q)

or

¬(P ∧ Q) ⇔ (¬P) ∨ (¬Q)

500

The set of complex numbers with an imaginary part

{zεC: z NOT ∈ R}

500

How do you show set A is equal to set B?

First show A ⊆ B. Then show B ⊆ A.

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