I am not able to understand the concepts of cache inclusion property in multi-level caching. As per my understanding, if we have 2 levels of cache, L1 and L2 then the contents of L1 must be a subset of L2. This implies that L2 must be at least as large as L1. Further, when a block in L1 is modified, we have to update in two places L2 and Memory. Are these concepts correct ?
In general, we can say adding more levels of cache is adding more levels of access in memory hierarchy. Its always trade-off between access time and latency. larger the cache, more we can store, but takes more time to search through. As you have said, L2 cache must be larger than L1 cache. otherwise its failing the basic purpose of the same.
Now coming to whether L1 a subset of L2. Its not always necessary. There is Inclusive cache hierarchy and exclusive cache hierarchy. In inclusive, as you said the last level is superset of all other caches.
you can check this presentation for more details PPT.
Now updating different levels, is a cache coherence problem & larger the number of levels, larger the headache. You can check various protocols here: cache coherence