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give chase (to someone or something)
to chase someone or something. The dogs gave chase to the fox. A mouse ran by, but the cat was too tired to give chase.
Pursue, as in The police gave chase to the robber. [c. 1700]
give ˈchasebegin to run after somebody/something in order to catch them: We gave chase along the footpath.
To engage in pursuit of quarry: Police gave chase to the speeding car.