Which of the following statements are true, and which are false? Give reasons for your answers in the form of a short proof or a counterexample. i) There are at least two ways of describing the set ,8,7{ . ...}
ii) Any function with domain R×R is a binary operation. iii) The graph of every function from ]1,0[ to R is infinite. iv) The function :f R → R , defined by )x(f = x x , is an odd function. v) The domain of the function ,g f o where )x(f = x and )x(g = 2 − ,x is ]2, ]−
Answers
Which of the following statements are true, and which are false? Give reasons for your answers in the form of a short proof or a counterexample.
i) There are at least two ways of describing the set ,8,7{ . ...}
ii) Any function with domain R×R is a binary operation.
iii) The graph of every function from ]1,0[ to R is infinite.
iv) The function :f R → R , defined by )x(f = x x , is an odd function.
v) The domain of the function ,g f o where )x(f = x and )x(g = 2 − ,x is ]2, ]−
Step-by-step explanation:
Which of the following statements are true, and which are false? Give reasons for your answers in the form of a short proof or a counterexample.
i) There are at least two ways of describing the set ,8,7{ . ...}
True:
(a) Integers greater than or equal to 7, (b) \{a_n|a_n = a_{n-1}+1\}, a_0 = 7, n = 1,2,...\\{a ₐ ∣aₐ =a ₐ₋₁ +1 },a₀ =7,n=1,2,...
ii) Any function with domain R×R is a binary operation.
True:
A binary operation on a set S can be defined as a mapping of the elements of the Cartesian product S * S to S. Taking R as S and see that the statement is true.
iii) The graph of every function from ]1,0[ to R is infinite.
False:
A counter example: y = x² which is finite at any point of [0,1].
iv) The function :f R → R , defined by )x(f = x x , is an odd function.
True:
Consider f(-x) = (-x)|-x| = -x|x| = -f(x)f(−x)=(−x)∣−x∣=−x∣x∣=−f(x). Therefore, by definition, f(x) is an odd function.
v) The domain of the function ,g f o where )x(f = x and )x(g = 2 − ,x is ]2, ]−
True: The domain of g(x) is (-infinity,2] and the domain of f(x) is x>0, which holds automatically for any x, as far as g(x) is always greater than zero. Therefore, the domain of the result is (-infinity,2] .
Step-by-step explanation:
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