Relations and Functions

**Cartesian product of two sets:**Two non-empty sets P and Q are given. The Cartesian product $\mathrm{P}\times \mathrm{Q}$ is the set of all ordered pairs of elements from P and Q, i.e.,

P × Q = {(p, q) : p ∈ P and q ∈** Q}**

**Example:** If P = {x, y} and Q = {-1, 1, 0}, then $\mathrm{P}\times \mathrm{Q}$ = {(x, -1), (x, 1), (x, 0), (y, -1), (y, 1), (y, 0)}

If either P or Q is a null set, then P × Q will also be a null set, i.e., $\mathrm{P}\times \mathrm{Q}$ = .

In general, if A is any set, then A × .

**Property of Cartesian product of two sets:**- If n(A) = p, n(B) = q, then n(A × B) = pq.
- If A and B are non-empty sets and either A or B is an infinite set, then so is the case with A × B
- .A × A × A = {(a, b, c) : a, b, c ∈ A}. Here, (a, b, c) is called an ordered triplet.
- A × (B ∩ C) = (A × B) ∩ (A × C)
- A × (B ∪ C) = (A × B) ∪ (A × C)

- Two ordered pairs are equal if and only if the corresponding first elements are equal and …

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