Unit 5 sets, logic and probability Review Sheet – TEST on March 19th !
What you need to know:
Sets:
*types of number sets (natural numbers, integers, rational, irrational, and real numbers) and the
correct notation (complement, number of elements, element, subset, intersection, union)
*how to set up and solve Venn Diagrams
logic:
*know how to write and interpret a conjuction, disjunction, and exclusive disjunction
*how to set up and complete a truth table
*know the terminology (tautology, logical contradiction, logically equivalent, implication, converse,
inverse, contrapositive)
probability:
* how to calculate probability with independent and dependent events
*create and calculate probability from a tree diagram
*calculate conditional probability “given that” in Venn diagram, table, or tree diagram
*understand the meaning of mutually exclusive/disjoint
*to prove two events A and B are independent, show: P( A  B)  P( A)  P( B)
*The union of probabilities can be found by the following: P( A  B)  P( A)  P( B)  P( A  B)
Part I: sets
1.
Determine whether A⊂B: a.) A = {2, 4, 6, 8} and B = {x⃒ 0 < x < 10, x∈ ℤ}
b.) A = {x⃒ 2 < x ≤ 4, x ∈ ℚ} and B = {x⃒ 0 ≤ x < 4, x∈ ℝ }
2.
Shade the appropriate areas for:
a.) N’
3.
b.) M ⋂ N
c.) M ⋂ N’
Consider the sets U = {x⃒ x ≤ 60, x ∈ ℤ+ } , A = {factors of 60}, and B = {factors of 30}
a. List the sets of A and B.
b. Illustrate sets A and B on a Venn Diagram. Be sure to write the elements in the appropriate places.
4. All attendees of a camp left something at home. 11 forgot to bring their towel, and 23 forgot their hat. Of the
30 campers, how many had neither a hat nor a towel?
Part II: logic
5.
Consider the propositions p: cakes are sweet and q: cakes are full of sultanas.
Write each of the following using logic symbols:
a.
If cakes are not sweet, they are not full of sultanas
b. If cakes are not sweet, they are full of sultanas.
c. Cakes are full of sultanas and they are not sweet.
d. Cakes are not sweet or they are full of sultanas.
6.
In what case can you write the symbol ⟺? Write two propositions that would you would be able to use the
equivalence symbol ⟺ , write your statement using symbols and words.
7.
Represent the truth sets of the following on Venn Diagram:
a.) p ⋁ q
b.) ∽ ( 𝑝 ∧ 𝑞 ∧ 𝑟)
8.
Construct a truth table for the following and state whether the statement is a tautology, logical contradiction or
neither. Are parts a) and b) logically equivalent? How do you know?
a.) (𝑝 ∧ 𝑞) ∧ ~(𝑝 ∨ 𝑞)
b.) ( 𝑝 ⋁ ~ 𝑞) ⇒ 𝑞
9.
Write down, in words, the inverse, converse and contrapositive for the implication:
“The diagonals of a rhombus are equal.”
Part III: probability
10. What is meant by a.) independent events?
b.) disjoint events?
11. Given P(Y) = 0.35 and P(X ∪ Y) = 0.8, and that X and Y are mutually exclusive events, find:
a.) P(X ∩ Y)
b.) P(X)
c.) the probability that X occurs or Y occurs, but not both X and Y.
12. Divers A, B and C have a 10%, 20%, and 30% chance of successfully finding an artifact in a shipwreck. If they all
try independently of one another, what is the probability that the artifact is found by at least one of the divers?
13. The probability that a man will be alive in 25 years is 3/5, and the probability that his wife will be alive is 2/3.
Determine the probability that in 25 years:
a.) Both will be alive
b.) At least one will be alive
c.) only the wife will be alive
14. Two cards are pulled from a standard deck without replacement. What are the chances of getting a heart and a
club in any order?
15. If P(X) = 1/4, P(Y’) = 1/6 and P(X ∪ Y) = 7/8, are X and Y independent?
16. An automatic gate has a 95% chance of working on any particular day. Find the probability that it will be
working on at least one of the next two days.
17. On any day, the probability that a boy eats his prepared lunch is 0.5. The probability that his sister eats her
lunch is 0.6. The probability that the girl eats her lunch given that the boy eats his is 0.9. Draw a tree diagram
representing the situation. Determine the probability that:
a.) both eat their lunch
b.) the boy eats his lunch given that the girl eats hers
18. In a team of 30 judo players, 13 have won a match by throwing (T), 12 have won by hold-down (H), and 13 have
won by points decision (P). 2 have won matches by all three methods. 5 have won matches by throwing and
hold-down. 4 have won matches by hold-down and points decision. 3 have won matches by throwing and
points decision.
a.) Draw a Venn diagram to display this information.
b.) Find:
i. P(T ⋂ H)
ii. P(P)
iii. P(H⃒P)
iv. P(T ∪ P)
v. P(T⃒H’)
vi. P((T∩P)⃒H)
Part Iv: ib practice!
1.
Two propositions p and q are defined as follows.
p: Jones passed this course
q: Smith passed this course
(a)
(b)
Write in symbolic form
(i)
neither Jones nor Smith passed the course;
(ii)
it is not the case that Jones and Smith both passed the course.
Complete the following truth table for the logic statement  p  q.
p
q
T
T
T
F
F
T
F
F
p
pq
(Total 4 marks)
2.
Given
the set of integers,
the set of rational numbers,
(a)
Write down an element that belongs to

.
(b)
Write down an element that belongs to

.
(c)
Write down an element that belongs to
.
(d)
Use a Venn diagram to represent the sets
,
and
the set of real numbers.
.
3. In a survey of 52 students it was found that 30 study Spanish and 15 have computers. Seven of the students who
study Spanish also have computers.
(a)
Copy and complete this table.
Study
Spanish
Do not study
Spanish
Total
Have computers
Do not have computers
Total
52
(3)
(b)
Draw and label fully a Venn diagram to illustrate this information. Use U to represent the set of all
students surveyed, S the set of students who study Spanish and C the set of students who have
computers.
(2)
(c)
Describe, in words, the set represented by C  S .
(2)
(d)
Find n(C  S ).
(1)
A student is selected at random to attend a computer workshop given in Spanish.
(e)
What is the probability that the student
(i)
has a computer and studies Spanish?
(2)
(ii)
as a computer but does not study Spanish?
(2)
(iii)
as a computer if he/she studies Spanish?
(2)
(Total 14 marks)
4.
On a particular day 100 children are asked to make a note of what they drank that day.
They are given three choices: water (W), coffee (C) or fruit juice (F)
1 child drank only water.
6 children drank only coffee.
8 children drank only fruit juice.
5 children drank all three.
7 children drank water and coffee only.
53 children drank coffee and fruit juice only.
18 children drank water and fruit juice only.
(a)
Represent the above information on a Venn Diagram.
(4)
(b)
How many children drank none of the above?
(2)
(c)
A child is chosen at random. Find the probability that the child drank
(i)
coffee;
(ii)
water or fruit juice but not coffee;
(iii)
no fruit juice, given that the child did drink water.
(4)
(d)
Two children are chosen at random. Find the probability that both children drank all three choices.
(3)
(Total 13 marks)
5.
The probability, p, that James gets up before 07.00 is 0.95.
If James gets up before 07.00, the probability, t, that he arrives at school on time is 0.98.
If James gets up later than 07.00, the probability that he arrives at school on time is 0.55.
The above information is represented by the following tree diagram.
0.98
0.95
t
p
not p
0.55
not t
t
not t
(a)
Complete the tree diagram.
(b)
Calculate the probability that James gets up before 07.00 and is on time for school.
(c)
Calculate the probability that James does not arrive at school on time.
(Total 8 marks)