Master the significance of ordered pairs in CBSE Class 11 Applied Mathematics. Learn why (a, b) ≠ (b, a) with solved examples and practice problems.
An ordered pair is a pair of elements (a, b) where the order of elements is significant. Unlike a set {a, b} where {a, b} = {b, a}, an ordered pair (a, b) is not equal to (b, a) unless a = b. This concept is fundamental in defining Cartesian products and coordinate geometry.
Step 1: Given (x + 1, y - 2) = (3, 1).
Step 2: Equate corresponding components: x + 1 = 3 and y - 2 = 1.
Step 3: Solve for x and y: x = 2, y = 3.
Answer: x = 2, y = 3
The position of an element within an ordered pair dictates its role, such as representing an x-coordinate and a y-coordinate on a plane. If the order is swapped, the location or meaning of the pair changes entirely. This distinction allows us to map unique relationships between two distinct sets.
Step 1: Consider the pair (2, 5).
Step 2: Compare with (5, 2).
Step 3: Since 2 ≠ 5, the first elements are different, thus (2, 5) ≠ (5, 2).
Answer: The pairs are distinct.
The Cartesian product A × B is the set of all possible ordered pairs (a, b) such that a ∈ A and b ∈ B. The significance of arrangement ensures that every element of A is paired with every element of B in a specific sequence. This structure is the basis for defining functions and relations.
Step 1: Let A = {1, 2} and B = {3, 4}.
Step 2: Form pairs (a, b) taking a from A and b from B.
Step 3: A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.
Answer: {(1, 3), (1, 4), (2, 3), (2, 4)}