Discrete Math Nested Quantifiers Exercise
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Let T(x,y) mean that student x likes cuisine y, where the domain for x
consists of all students at your school and the domain for y consists of
all cuisines. Express each of these statements by a simple English
sentence.
a) ¬T(Abdallah Hussein,Japanese)
b)
∃
x T(x,Korean)
∧∀
x T(x,Mexican)
c)
∃
y (T(Monique Arsenault,y)
∨
T (Jay Johnson, y))
d)
∀
x
∀
z
∃
y((x̸=z)→¬(T(x,y)
∧
T(z,y)))
e)
∃
x
∃
z
∀
y(T(x,y)↔T(z,y))
f)
∀
x
∀
z
∃
y(T (x, y) ↔ T (z, y))
S
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i
o
n
a) Abdallah Hussein does not like Japanese cuisine.
b) Note that this is the conjunction of two separate quantified statements.
Some student at your school likes Korean cuisine, and everyone at your
school likes Mexican cuisine.
c) There is some cuisine that either Monique Arsenault or Jay Johnson
likes.
d) Formally this says that for every
x
and
z,
there exists a
y
such that if
x
and
z
are not equal, then it is not the case that both
x
and
z
like
y.
In
simple English, this says that for every pair of distinct students at your
school, there is some cuisine that at least one them does not like.
e) There are two students at your school who have exactly the same tastes
(i.e., they like exactly the same cuisines).
f)
For every pair of students at your school, there is some cuisine about
which they have the same opinion (either they both like it or they both do
not like it).
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Exercise involving nested quantifiers to express statements about students liking different cuisines at school, with solutions provided in simple English.
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Presentation Transcript
Discrete Math: Nested Quantifiers Exercise 4
Exercise Let T(x,y) mean that student x likes cuisine y, where the domain for x consists of all students at your school and the domain for y consists of all cuisines. Express each of these statements by a simple English sentence. a) T(Abdallah Hussein,Japanese) b) x T(x,Korean) x T(x,Mexican) c) y (T(Monique Arsenault,y) T (Jay Johnson, y)) d) x z y((x =z) (T(x,y) T(z,y))) e) x z y(T(x,y) T(z,y)) f) x z y(T (x, y) T (z, y))
Solution a) b) Note that this is the conjunction of two separate quantified statements. Some student at your school likes Korean cuisine, and everyone at your school likes Mexican cuisine. Abdallah Hussein does not like Japanese cuisine. c) There is some cuisine that either Monique Arsenault or Jay Johnson likes. d) Formally this says that for every x and z, there exists a y such that if x and z are not equal, then it is not the case that both x and z like y. In simple English, this says that for every pair of distinct students at your school, there is some cuisine that at least one them does not like. e) There are two students at your school who have exactly the same tastes (i.e., they like exactly the same cuisines). f) For every pair of students at your school, there is some cuisine about which they have the same opinion (either they both like it or they both do not like it).
References Discrete Mathematics and Its Applications, McGraw-Hill; 7th edition (June 26, 2006). Kenneth Rosen Discrete Mathematics An Open Introduction, 2nd edition. Oscar Le in A Short Course in Discrete Mathematics, 01 Dec 2004, Edward Bender & S. Gill Williamson
