Discrete Structures Week 6 Quiz 2

Question 1

definition of set

Answer 1

an unordered list of elements without repetitions

Question 2

set of S

Answer 2

S = { x C- DOMAIN/SET | P(x) }

for any x in DOMAIN/SET such that P(x) is true

Question 3

subset

Answer 3

A c_ B
A is a subset of B if every element of A is also an element of B

Question 4

empty set and S

Answer 4

empty set - set for which Po/(x) is a contradiction (always false)
V- x, Po/(x) = F

theorem
for any set S, empty c_S and S c_ S

1. S c_ S
   - we need to show that if V- x, x C- S then x C- S is always true so S c_ S
2. empty c_ S
   -want to show if x C- empty, then x C- S, in other words V- x, the conditional
   x C- empty –> x C- S is vacuously true

Question 5

induction

Answer 5

P(n), n >= 1, n C- DOMAIN

1. P(1 or base case) = T
2. P(k) -> P(k+1), k >= 1, k C- DOMAIN

then P(n) = T, V-n >= 1, n C- DOMAIN

Question 6

venn diagrams and operation of sets

Answer 6

intersection
A n B
= {x C- U | P(x) ^ Q(x) }

union
A u B
={x C- U | P(x) v Q(x)}

compliment
A^c = A - (bar on top)
= {x C- U | !P(x)}

disjoint
A n B = o/ empty set

Question 7

symbols for set of integers, natural numbers, and rational numbers, and real numbers

Answer 7

set of integers = Z
-numbers positive and negative whole numbers

set of all natural numbers = N
1 to inf

set of rational numbers = Q
- decimals and fractions

set of all real numbers = N
- everything

natural numbers c_ whole numbers c_ integers c_ rational numbers c_ real numbers

Question 8

when two sets are equal

Answer 8

A = B iff P(x) = (3 lines_ Q(x), V- x C- U

P iff Q, P-> Q and Q-> P

Question 9

symmetric difference

Answer 9

Xor
o with + inside

A Xor B
= {x C- U | P(x) Xor Q(X)}
= (A \ B) U (B \ A)

Question 10

set difference

Answer 10

A \ B
= {x C- U | P(x) ^ !Q(x)}