Show that the relation R defined in the set A of all triangles as R = {(T1, T2): T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?
R = {(T1, T2): T1 is similar to T2}
R is reflexive since every triangle is similar to itself.
Further, if (T1, T2) ∈ R, then T1 is similar to T2.
⇒ T2 is similar to T1.
⇒ (T2, T1) ∈R
∴R is symmetric
Now,
Let (T1, T2), (T2, T3) ∈ R.
⇒ T1 is similar to T2 and T2 is similar to T3.
⇒ T1 is similar to T3.
⇒ (T1, T3) ∈ R
∴ R is transitive.
Thus, R is an equivalence relation.
Now, we can observe that:
\(\frac {3} {6}\)=\(\frac {4} {8}\)= \(\frac {5} {10}\)= \(\bigg(\frac{1}{2}\bigg)\)
∴The corresponding sides of triangles T1 and T3 are in the same ratio.
Then, triangle T1 is similar to triangle T3.
Hence, T1 is related to T3.
Read the following text carefully:
Union Food and Consumer Affairs Minister said that the Central Government has taken many proactive steps in the past few years to control retail prices of food items. He said that the government aims to keep inflation under control without compromising the country’s economic growth. Retail inflation inched up to a three-month high of 5.55% in November 2023 driven by higher food prices. Inflation has been declining since August 2023, when it touched 6.83%. 140 new price monitoring centres had been set up by the Central Government to keep a close watch on wholesale and retail prices of essential commodities. The Government has banned the export of many food items like wheat, broken rice, non-basmati white rice, onions etc. It has also reduced import duties on edible oils and pulses to boost domestic supply and control price rise. On the basis of the given text and common understanding,
answer the following questions:
Relation is said to be empty relation if no element of set X is related or mapped to any element of X i.e, R = Φ.
A relation R in a set, say A is a universal relation if each element of A is related to every element of A.
R = A × A.
Every element of set A is related to itself only then the relation is identity relation.
Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R-1 is said to be an Inverse relation if R-1 from set B to A is denoted by R-1
If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.
A relation R is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.
A relation is said to be transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A
A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive.