Final answer:
In probability theory, the probability of the intersection of two events is always less than or equal to the probability of either event, while the probability of their union is bounded by the sum of individual probabilities, less the probability of the intersection to eliminate any double-counting.
Step-by-step explanation:
The question pertains to probability theory in mathematics and involves understanding the relationship between the probabilities of individual events and their intersections and unions. To show the inequalities p(c1 ∩ c2) ≤ p(c1) ≤ p(c1 ∪ c2) and that p(c1 ∪ c2) ≤ p(c1) + p(c2), one must apply basic principles of probability.
Firstly, the intersection of two sets, c1 ∩ c2, represents outcomes that are common to both c1 and c2, therefore its probability will always be less than or equal to the probability of either subset alone.
Secondly, the union of two sets, c1 ∪ c2, combines all unique outcomes from both subsets. Its probability is bounded below by the probability of the larger subset and bounded above by the sum of the probabilities of the two subsets, since some outcomes might be double-counted. The probability of the union is less than or equal to the sum of individual probabilities, minus the probability of the intersection to correct for over-counting.