Question

Let $\xi $= the set of all triangle, P = the set of all isosceles triangle, Q = the set of all equilateral triangle, R = set of all right angled triangle. What do the set \[P\cap Q\text{ and R}-\text{P}\] represent?
A. The set of isosceles triangle; the set of non-isosceles right angled triangle.
B. The set of isosceles triangle; the set of right angled triangle.
C. The set of equilateral triangle; the set of non-isosceles right angled triangle.
D. The set of isosceles triangle; the set of equilateral triangle.

Answer 1

Hint:We are given a different set of triangles denoted as P, R and Q. We are asked to find \[P\cap Q\text{ and R}-\text{P}\], we will work on each of them one by one. We know $\cap $ is called intersection, so \[P\cap Q\] is defined as the intersection of P and Q, it contains all those elements which are common to P as well as to Q. So, we will look what is common in P and Q both.
Then, we will work on R - P, we know '-' means subtraction, R - P will be defined as the set of remaining elements of R after separating elements of P from it, we will separate P from R and look what is left into set R and then we will get our required solution.

Complete step by step answer:
We are given that,
$\xi $= the set of all triangle,
P = the set of all isosceles triangle,
Q = the set of all equilateral triangle,
R = set of all right-angled triangle
We are asked to find what does \[P\cap Q\text{ and R}-\text{P}\] represent.
Firstly, we will learn that what does $\cap $ mean, $\cap $ is defined as an intersection for any set A and B, $A\cap B$ is defined as A intersection B, it contains all those elements which lie in A as well as in B. Means $A\cap B$ contain all those thing which are common to both A and B.
So for \[P\cap Q\] it is defined as the set which is common to P as well as Q.
We have P as set of all isosceles triangle while Q denotes the set of all equilateral triangle.
By definition, we know the isosceles triangle are those whose 2 sides are equal, as the equilateral triangle has all sides equal. Means their 2 sides are also equal. So, all equilateral triangle is isosceles.
So, we get that in P and Q, the equilateral triangle is common. So, \[P\cap Q\] will denote the set of all equilateral triangle.
Now, we will work on R - P.
Firstly, we will define '-' means. '-' is the sign of subtraction for any set say A and B. A - B means, just subtract the elements of B that lie in A. So, A - B is defined as set A which has no elements of B lying into it. So,
R - P is defined as those elements of R which contain all elements of P from it.
We have R as set of all right-angled triangle while P as set of all isosceles triangle.
We know, the right-angled triangle can be isosceles or non-isosceles if we subtract set of the isosceles triangle from R. We are just left with the right-angled triangle which are not isosceles.
So, R - P is the set of all non-isosceles right-angled triangle.
So, we get that, the correct option is C.

Note:
Remember that, the isosceles definition says, if any two sides are equal in any triangle then it will be an isosceles triangle, if all sides are equal then also the definition is true and that’s why all equilateral triangle is also an isosceles triangle. The converse is not true. While finding R-P we have to just look at what is left in R after removing elements of P from it. We do not have to look in P after subtraction.