$\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Overcoming Wrong Intuition This is a widespread intuitive argument, asserting that $\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$:
Between any two rational numbers there's an irrational one and vice versa. So $\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$.
How can one convince the learner that this argument is invalid?

It is natural to have such misconceptions, and it is also hard for someone else to help you get rid of them. You use the right concepts instead repeatedly so that after sometime your wrong intuition seems absurd to you. As von Neumann says "In mathematics, you don't understand things. You just get used to them." That is very true. I have been there.
