Tautologies and Contradictions
What is a Tautology?
A statement that is always true.
Rewrite the statement using a quantifier.
Some hotdogs have mustard.
∃ hotdogs have mustard.
Given the statement "Quin is running."
What is the term?
Quin
20 students took a survey.
5 like math and science
12 like science only
1 likes neither
How many students like only math?
2 students
Given the sets:
A={1,5,7,8}
B={2,5,8}
What is A ∩ B?
{5,8}
What is a Contradiction?
A statement that is always False.
Rewrite the statement using a quantifier.
All of the desks are grey.
∀ of the desks are grey.
Given the statement "The dog is brown."
What is the predicate?
is brown
40 students took a survey.
12 like kickball only
13 like jump rope only
10 like neither
How many students like kickball and jump rope?
5 students
Given the sets:
A={1,5,9,4}
B={3,7,2,9}
What is A ∩ B?
∅
(empty set)
Is the following a tautology, contradiction, or neither?
p ∧ ~p
Contradiction
Rewrite the statement using a quantifier.
Some cats are orange but all cats have tails.
∃ cats are orange but ∀ cats have tails.
Given the statement "x is even."
What is the term?
x
30 students took a survey
11 play guitar
12 play piano
0 play both
How many students play neither?
7 students
What is the rule of the set?
F={0,1,1,2,3,5,8,13,...}
Fibonacci Sequence
(add the previous 2 numbers together)
Is the following a tautology, contradiction, or neither?
(p → q) ∨ (q → p)
tautology
What are the 2 meanings of ∀ ?
All
For all
Given the statement "Alan is walking his dog."
What is the predicate?
260 students took a survey
100 like red
120 like blue
30 like green and blue
50 like red and green
10 like all 3 colors
70 like green only
40 like red only
50 people like blue
20 people like red and blue but not green
20 like none of the colors
How many students like red and blue?
30 students
Given the sets:
A={1,8,3,9}
B={7,2,9,3}
C={1,5,9,2}
What is A ∪ (B ∩ C)?
{1,2,3,8,9}
Is the following a tautology, contradiction, or neither?
(p ∧ q) ∨ (~p ∨ (p ∧ ~q))
tautology
What are the 2 meanings of ∃?
Some
There exists
Given the statement "Some dogs are not beagles." What is the first predicate?
is a dog
260 students took a survey
100 like red
120 like blue
30 like green and blue
50 like red and green
10 like all 3 colors
70 like green only
40 like red only
50 people like blue
20 people like red and blue but not green
20 like none of the colors
How many students like exactly 2 of the colors?
50 students
Given the sets:
A={1,3,7,9,0}
B={1,2,6,9,3,8}
C={1,8,4,2,0}
D={1,6,3,8,9,0}
What is {A ∩ B} ∪ {C ∩ D}?
{0,1,3,8,9}