Relations

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An ordered pair is a pair of objects taken in a specific order, written as (a, b), where 'a' is the first member and 'b' is the second member. The order is significant. Unlike sets where {a, b} = {b, …

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Two ordered pairs (a, b) and (c, d) are equal if and only if a = c and b = d. Written as: (a, b) = (c, d) ⟺ a = c and b = d. Example: If (2x, y - 3) = (2, 1), then 2x = 2 and y - 3 = 1, giving x = 1 a…

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The Cartesian product of sets A and B, written as A × B, is the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B. Formally: A × B = {(a, b) : a ∈ A, b ∈ B}. Example: If A = {1, 2} and B = {x,…

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If n(A) = 3 and n(B) = 4, then n(A × B) = n(A) × n(B) = 3 × 4 = 12 elements. In general, n(A × B) = n(A) × n(B). This is because each element of A can be paired with each element of B.

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No, in general A × B ≠ B × A. The Cartesian products are equal only when A = B. Example: If A = {1, 2} and B = {a, b}, then A × B = {(1, a), (1, b), (2, a), (2, b)} while B × A = {(a, 1), (a, 2), (b, …

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A relation R from a non-empty set A to a non-empty set B is defined as a subset of the Cartesian product A × B. If (a, b) ∈ R, we say 'a is R-related to b' and write it as aRb. The element b is called…

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For a relation R from set A to set B: • Domain: Set of all first elements of ordered pairs in R • Range: Set of all second elements of ordered pairs in R • Co-domain: The entire set B Note: Range ⊆ C…

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The total number of relations from A to B is 2^(pq). This is because a relation is a subset of A × B, and A × B has pq elements. The number of subsets of a set with pq elements is 2^(pq). Example: If …