Consumer choice: How could a consumer weakly prefer A to B *and* B to A? From Perloff, Microecomics, 6th edition (Pg. 75): 

A consumer chooses between bundles of goods by ranking them as to the pleasure
the consumer gets from consuming each. We summarize a consumer’s ranking using
a preference relation, such as the consumer weakly prefers Bundle a to Bundle b, if the consumer likes Bundle a at least as much as Bundle b.

Given this weak preference relation, we can derive two other relations. If the consumer weakly prefers Bundle a to b, but the consumer does not weakly prefer
b to a, then we say that the consumer strictly prefers a to b—would definitely
choose a rather than b if given a choice—which we write as a b.

If the consumer weakly prefers a to b and weakly prefers b to a —— then we say that the consumer is indifferent between the bundles, or likes the
two bundles equally.



My question is: When you have declared that a consumer weakly prefers A to B, how could you then argue that a consumer could also prefer (weakly) B to A? Aren't they mutually exclusive?
No, weak orderings are not mutually exclusive, but strict orderings are, exactly as for the usual comparison of real numbers: you can have $x \geq y$ and $y \geq x$ (if $x=y$), but $x&gt;y$ and $y&gt;x$ are incompatible with each other.

The weak orderings $A \succeq B$ and $B \succeq A$ mean that the decision-maker is indifferent between $A$ and $B$: he might therefore choose any of these options indifferently.