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Already read: $\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable?

I am learning logic for the first time, about six months after finishing my undergraduate degree. I notice that there seem to be some similarities between set theory and logic.

For example, if $A$ is a set, $a$ is a statement form, and $\mathbf{t}$ and $\mathbf{c}$ are a tautology and contradiction respectively, letting $U$ be a universal set [yes, I know this leads to a paradox] and $\varnothing$ be the empty set, I find that $U$ and $\mathbf{t}$ have similar properties, as do $\mathbf{c}$ and $\varnothing$.

Is my hunch wrong? For example, $A \cap \varnothing = \varnothing$, and $a \wedge \mathbf{c} = \mathbf{c}$.