[Math] Difference between $⊂$ and $⊆$

discrete mathematicselementary-set-theorynotation

I was solving the exercises in Discrete Mathematics and its applications book.

Determine whether each of these statements is true or
false.

  1. {0} ⊂ {0}
  2. {∅} ⊆ {∅}

I thought both 1 and 2 are true, but when I checked the answers I found that 1 is false and 2 is true.
I got confused and distracted because I don't know the difference between them.

Best Answer

If the book distinguishes between $\subset$ and $\subseteq$, then most likely the former symbol denotes proper inclusion, so $\{0\}\subset\{0\}$ is false. The latter symbol instead will denote inclusion (with possible equality).

However it's very common to find $\subset$ denoting inclusion (with possible equality), so one always has to check or try and infer from the context. Don't take Wikipedia pages as revealed truth.

It's so common that $\subset$ denotes nonstrict inclusion that somebody uses $\subsetneq$ to denote proper inclusion, for safety.

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