Question

Let \[{F_1}\] be the set of parallelograms, \[{F_2}\] be the set of rectangles, \[{F_3}\] be the set of rhombuses, \[{F_4}\] be the set of squares and \[{F_5}\] be the set of trapeziums in a plane. Then \[{F_1} \cup {F_2} \cup F{ \cup _3}{F_4}\]is a subset of what?
A) \[{F_1}\]
B) \[{F_2}\]
C) \[{F_3}\]
D) \[{F_4}\]
 E) \[{F_5}\]

Answer 1

Hint: Several sets are given in the problem. These sets contain types of geometrical shapes. Using the very basic properties of geometry, we will relate them among themselves, which leads us to the solution.

Complete step-by-step answer:
It is given that,
\[{F_1}\] is the set of parallelograms
\[{F_2}\] is the set of rectangles
\[{F_3}\] is the set of rhombuses
\[{F_4}\] is the set of squares, and
\[{F_5}\] is the set of trapeziums

By definition and property, we know that a parallelogram has two pairs of opposite sides that are parallel and equal in length.
Rectangle has two pairs of opposite sides. So a rectangle is also a parallelogram with four right angles.
Rhombus is a parallelogram with four equal sides.
Square has two pairs of opposite sides and all of them are equal. And square is a degenerate version of a rectangle.
From the above observation, we can say that all the rectangles, rhombuses and squares are parallelograms.
Hence we have,
\[
  {F_2} \subset {F_1}. \\
  {F_3} \subset {F_1}. \\
  {F_4} \subset {F_1}. \\
 \]
Therefore, \[{F_1} \cup {F_2} \cup F{ \cup _3}{F_4}\] \[ \subset {F_1}.\]

Hence we have \[{F_1} \cup {F_2} \cup F{ \cup _3}{F_4}\] is the subset of the set \[{F_1}\].

Note:
A set A is a subset of another set B if all the elements of the set A are the elements of the set B. If A is a set that is a subset of a set C and B is a set that is also a subset of the set C then $A \cup B$ is also a subset of the set C.