Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 1 / b

False. Take the sets $A=\{1\}$, $B=\varnothing$ and $C=\varnothing$. Both intersections with $C$ are empty, although $A\ne B$. Intersecting with $C$ loses information about membership outside $C$.

False. Take the sets $A=\{1\}$, $B=\varnothing$ and $C=\{1\}$. Both unions with $C$ equal $\{1\}$, although $A\ne B$. Taking a union with $C$ conceals which of its elements belonged to the original set.

True. The two operations retain complementary membership information. For $x\in C$, equality of the intersections makes $x\in A$ equivalent to $x\in B$. For $x\notin C$, equality of the unions makes $x\in A$ equivalent to $x\in B$. These cases exhaust every possible element. Set extensionality therefore gives $A=B$. Equivalently, reconstruct either set fromSubstituting the two given equalities into this expression gives $B$.

False. With the sets $A=\{1\}$, $B=\varnothing$ and $C=\{1\}$, both set differences are empty but $A\ne B$. Removing $C$ loses the distinction between their memberships inside $C$.