Types of relations
Q. Consider the relation R on the set of real numbers defined by R = {(x, y) | x^2 + y^2 = 1}. What type of relation is R?
A.
Reflexive
B.
Symmetric
C.
Transitive
D.
None of the above
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Solution
R is symmetric because if (x,y) is in R, then (y,x) is also in R. It is not reflexive or transitive.
Correct Answer: B — Symmetric
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Q. If R is a relation on the set A = {1, 2, 3} defined by R = {(1, 2), (2, 3), (3, 1)}, which of the following properties does R possess?
A.
Reflexive
B.
Symmetric
C.
Transitive
D.
All of the above
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Solution
R is not reflexive as (1,1), (2,2), (3,3) are not in R. It is symmetric as (2,3) implies (3,2) is not in R. It is transitive as (1,2) and (2,3) implies (1,3) is not in R. Thus, R is not all of the above.
Correct Answer: C — Transitive
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Q. If R is a relation on the set {1, 2, 3, 4} defined by R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}, what type of relation is R?
A.
Reflexive
B.
Symmetric
C.
Transitive
D.
Both reflexive and symmetric
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Solution
R is reflexive because it contains all pairs (a, a) and symmetric because (1,2) implies (2,1).
Correct Answer: D — Both reflexive and symmetric
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Q. If R is a relation on the set {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}, which of the following is true?
A.
R is reflexive
B.
R is symmetric
C.
R is transitive
D.
Both 1 and 2
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Solution
R is reflexive and symmetric, but not transitive. Thus, both 1 and 2 are true.
Correct Answer: D — Both 1 and 2
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Q. If R is a relation on the set {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3), (1, 2)}, is R a partial order?
A.
Yes
B.
No
C.
Only reflexive
D.
Only transitive
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Solution
R is not a partial order because it is not transitive; (1,2) and (2,2) do not imply (1,2).
Correct Answer: B — No
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Q. If R is a relation on the set {a, b, c} defined by R = {(a, b), (b, c)}, what can be said about R?
A.
Reflexive
B.
Symmetric
C.
Transitive
D.
None of the above
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Solution
R is neither reflexive, symmetric, nor transitive as it does not satisfy any of the properties.
Correct Answer: D — None of the above
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Q. If R is a relation on the set {a, b, c} defined by R = {(a, b), (b, c)}, which property does R NOT have?
A.
Reflexive
B.
Symmetric
C.
Transitive
D.
None of the above
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Solution
R is not symmetric as (b,c) does not imply (c,b) is in R. It is reflexive and transitive.
Correct Answer: B — Symmetric
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Q. If R is a relation on the set {x, y, z} defined by R = {(x, y), (y, z), (z, x)}, what can be said about R?
A.
Reflexive
B.
Symmetric
C.
Transitive
D.
None of the above
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Solution
R is neither reflexive, symmetric, nor transitive.
Correct Answer: D — None of the above
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Q. Let A = {1, 2, 3, 4} and R be the relation defined by R = {(a, b) | a < b}. How many ordered pairs are in R?
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Solution
The pairs are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, there are 6 ordered pairs.
Correct Answer: B — 6
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Q. Let A = {1, 2, 3, 4} and R be the relation defined by R = {(x, y) | x < y}. How many ordered pairs are in R?
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Solution
The ordered pairs are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, there are 6 ordered pairs.
Correct Answer: B — 6
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Q. Let R be a relation on the set of natural numbers defined by R = {(m, n) | m divides n}. Is R a partial order?
A.
Yes
B.
No
C.
Only reflexive
D.
Only transitive
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Solution
R is reflexive, antisymmetric, and transitive, thus it is a partial order.
Correct Answer: A — Yes
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Q. Which of the following relations is an equivalence relation on the set of integers?
A.
x ~ y if x + y is even
B.
x ~ y if x - y is prime
C.
x ~ y if x > y
D.
x ~ y if x = y
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Solution
The relation x ~ y if x + y is even is reflexive, symmetric, and transitive, thus it is an equivalence relation.
Correct Answer: A — x ~ y if x + y is even
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Q. Which of the following relations is not a function?
A.
R = {(1, 2), (2, 3), (3, 4)}
B.
R = {(1, 2), (1, 3)}
C.
R = {(2, 3), (3, 4)}
D.
R = {(4, 5), (5, 6)}
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Solution
R = {(1, 2), (1, 3)} is not a function because the input 1 maps to two different outputs.
Correct Answer: B — R = {(1, 2), (1, 3)}
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Q. Which of the following relations on the set of integers is not a function?
A.
R1 = {(1, 2), (1, 3)}
B.
R2 = {(2, 3), (3, 4)}
C.
R3 = {(4, 5)}
D.
R4 = {(5, 6), (6, 7)}
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Solution
R1 is not a function because the input 1 maps to two different outputs (2 and 3).
Correct Answer: A — R1 = {(1, 2), (1, 3)}
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