Prove that if the confidence of the rule {a} ?? {b} is less than the support of {b}, then: i. c({a} ?? {b}) > c({a} ?? {b}), ii. c({a} ?? {b}) > s({b}), where c(·) denote the rule confidence and s(·) denote the support of an itemset.

Suppose we have market basket data consisting of 100 transactions and 20
items. If the support for item a is 25%, the support for item b is 90% and the
support for itemset {a, b} is 20%. Let the support and confidence thresholds
be 10% and 60%, respectively.

is always positive because P({a})P({b}) > P({a, b}).