Verify whether the given set is an empty set or not.
(i) \[\left\{ x:x\text{ is a point common to any two parallel lines} \right\}\]
VerifiedHint:From the given condition, check if they contain elements or not. In an empty set there won’t be any value that will satisfy the condition. If there are values that satisfy the condition then they are not examples of empty sets.
Complete step-by-step answer:
The empty set is the unique set having no elements, its size or cardinality, which is the count of elements in the set is zero. We can ensure that the empty set exists by including an axiom of empty set. Now the empty set can be also referred to as a null set. However null set is a distinct notion within the context of measure theory.
Null or empty set can be also described as a set of measure zero. The common relations for the empty set include \['\left\{ {} \right\}'\] and \[\phi \].
Now for the given 2 conditions let us state whether they are examples of empty set or not.
From the given condition, it states that\[x\]is a point which is common to any two parallel lines. We know that parallel lines are lines in a plane which do not meet in, two straight lines in a plane that do not intersect at any point are said to be parallel.
From the figure \[{{l}_{1}}\]and \[{{l}_{2}}\] are two parallels. Since these 2 lines are parallel, they will never meet to cross at a single point. Hence no such point will exist which is common to both the lines.
Thus the given set is an example of an empty set.
Hence the given set is an empty set.
Note: We can give examples of empty sets as
(i) The set of humans living on the Moon, which is an empty set.
(ii) The set of 1000m tall people
(iii)The set of all Triangles with four sides.
