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Relation Definition

The relation ...

Question

The relation $R$ defined on the set $A = {1, 2, 3, 4, 5}$ as $R = {(a, b) : | a^{2} - b^{2} | < 16}$ has cardinality equal to

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Solution

$(a, b) \in R ⟺ | a^{2} - b^{2} | < 16$
$a = 1 \Rightarrow | 1 - b^{2} | < 16 \Rightarrow b = 1, 2, 3, 4$
$a = 2 \Rightarrow | 4 - b^{2} | < 16 \Rightarrow b = 1, 2, 3, 4$
$a = 3 \Rightarrow | 9 - b^{2} | < 16 \Rightarrow b = 1, 2, 3, 4$
$a = 4 \Rightarrow | 16 - b^{2} | < 16 \Rightarrow b = 1, 2, 3, 4, 5$
$a = 5 \Rightarrow | 25 - b^{2} | < 16 \Rightarrow b = 4, 5$
∴ Total number of elements $= 19$

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