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what do they mean by "in a weak interaction, quark flavor can change"

I know that in a weak interaction strangeness is conserved... so is this what the statement is referring to.

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    $\begingroup$ In weak interactions strangeness is not conserved, at least not in the charged currents. Indeed, a strange quark can turn e.g. into a up quark via the cabibbo angle mixing by exchanging a W boson. Flavor changing neutral currents are instead forbidden at tree level. $\endgroup$ – TwoBs Jun 11 '14 at 6:58
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Flavor means that distinct species of elementary particles may be distinguished within more general types; the Standard Model of Particle Physics describes six flavors of quarks and six flavors of leptons.

Since there are three quark species which equally carry electro-magnetic charge +2/3 as well as three quark species of charge -1/3 therefore the distinctiveness (within each charge variant, and also in general) is ultimately and practically that each quark species has a different mass.

Similarly, there are three species of leptons which all carry a charge of -1 and which are distinguished by their significantly different masses.

For the remaining three species of electrically neutral leptons (i.e. the species of neutrinos) a distinction by mass is at least not practical; therefore "neutrino flavor" is usually not taken in the sense of different mass, but instead referring to the distinctive coupling to the charged lepton flavors.

what do they mean by "in a weak interaction, quark flavor can change"

Any process of particle physics, as described by the Standard Model, by which the number of one particular quark species "$j$" changes (counting "$\#~\text{quarks}_j - \#~\text{anti-quarks}_j$" of that species "$j$"; and where the number of other quark species will have to change accordingly since overall the process should conserve baryon number:

$$\sum_{\text{all species}} \#~\text{quarks} - \sum_{\text{all species}} \#~\text{anti-quarks} =\text{(remains constant)}$$ ) is always due to weak interaction; and specificly, in detail, even always due to charged weak interaction, i.e. involving exchanges (or decays, or production) of $\mathbf W$ bosons (which are charged).

That means also that any other form of interaction (strong, electromagnetic, neutral weak, or gravitative) does not change the flavor (masses) of given quarks (but could at most create or annihilate quark anti-quark pairs; leaving the number of quarks per species constant).

I know that in a weak interaction strangeness is conserved..

Wrong; there are plenty charged weak processes known in which "strangeness" is not conserved, i.e. the number "$\#~\text{quarks}_{\text{strange}} - \#~\text{anti-quarks}_{\text{strange}}$" differs between initial and final state.

(Perhaps you misremebered the fact that the strong interaction conserves "strangeness", as well as separately any flavor.)

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LoLwhut? Weak interaction can change quark’s flavor arbitrarily. Practically, any quark is converted to “u” or “d” promptly (both variants happen, u⇌d included) namely by means of weak interaction.

It is strong interaction that conserves individual flavors and derived “numbers”, such as stangeness and isobaric spin. It is, in fact, the reason that makes hadronization of higher-generations quarks possible at all.

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