Q1T1
Let A = {3,5,7} and B = {15,16,17,18}
Write the Domain and co-domain
Domain - {3,5,7}
Co-Domain - {15,16,17,18}
p: Firemen work hard
q: Firemen wear red suspenders
Write p ⋀ q
Firemen work hard and Firemen wear red suspenders
p
therefore p or q
Generalization
23, 14, 5, -4, -13, ...
-22, -31, -40
Final Statement of an argument
(Q2T1)
Conclusion
A Variable is important in mathematics to...
Maintain generality and avoid ambiguity
Negate this statement:
-10 < x < 2
-10 ≥ x or x ≥ 2
If p then q
not q
therefore not p
Modus Tollens
Rewrite as a sum:
7
∑ (200 - a2)
a = 4
184 + 175 + 164 + 151
Assumptions or hypotheses of an argument
(Q2T1)
Premises
State whether these are True or False:
1) {2} ∈ {{1}, {2}}
2) {2} ⊆ {{1}, {2}}
1) True
2) False
Let p = "x < 3," q = "x = 3," and r = "4 < x"
4 < x ≤ 3
r ∧ (p ∨ q)
Is the following argument invalid because of converse or inverse error?
If John and Carl sit next to each other, then the classroom with be loud.
The classroom is loud.
Therefore John and Carl are sitting next to each other.
Converse Error
Evaluate These Expressions:
1) 15C2
2)
4
∏ 4x + 1
x = 2
1) 105
2) 1989
What is this an example of?:
The student body of 140 students wants to
elect a president, vice president, and
secretary.
Permutation
There is a positive integer that is less than or equal to every positive integer
Existential Universal Statement
It has opposite truth value from p: if p is true, ~p is false; if p is false, ~p is true.
Conjunction
Rewrite the following statement in if-then form without using the word "necessary": Getting an answer of 10 for problem 16 is a necessary condition for solving problem 16 correctly.
If someone does not get an answer of 10 for problem 16, then the person will not have solved problem 16 correctly.
Or: If someone solves problem 16 correctly, then the person got an answer of 10
16! / 13!4!
140
What is this an example of?:
Team A and Team B are playing in a tournament How many ways can the tournament end if a team has to win three games total or two games in a row?
Possibility Tree
Let A = {a,b,c} and B = {u,v}
(A x B) x A:
A x B
{(a,u),(a,v),(b,u),(b,v),(c,u),(c,v)}
Compose a Truth Table and state whether or not it is logically equivalent:
(p ∧ q) ∧ r and p ∧ (q ∧ r)
p q r|p ∧ q|q ∧ r|(p ∧ q) ∧ r|(p ∧ (q ∧ r)|(p ∧ q)
Then Compare: (p ∧ q) ∧ r with p ∧ (q ∧ r)
Logically Equivalent
Compose a Truth Table, tell which ones are CR, invalid or valid, and words explaining why:
If p or q then r
r ∨ q
∴ p ∨ q
p q r|r ∨ q|p ∨ q
1st Row (CR)
2nd Row
3rd Row (CR)
4th Row
5th Row (CR)
6th Row
7th Row (Invalid)
8th Row (CR)
It is invalid because the premises include 2 truths and the conclusion is false.
(n - 3)! / n!
1 / n(n - 1)(n - 2)
Solve the complex deduction:
The famous detective Percule Hoirot was called in to solve a baffling murder myster. He determined the following facts:
1) Lord Hazelton, the murdered man, was killed by a blow on the head with a brass candlestick.
2) Either Lady Hazelton or a maid, Sara, was in the dining room at the time of the murder.
3) If the cook was in the kitchen at the time of the murder, then the butler killed Lord Hazelton with a fatal dose of strychnine.
4) If Lady Hazelton was in the dining room at the time of the murder, then the chauffer killed Lord Hazelton.
5) If the cook was not in the kitchen at the time of the murder, then Sara was not in the dining room when the murder was committed.
6) If Sara was in the dining room at the time the murder was committed, then the wine steward killed Lord Hazelton.
Is is possible for the detective to deduce the identity of the murderer from these facts? If so, who did murder Lord Hazelton? (Assume there was only one cause of death.)
Yes, the killer was the chauffeur