  We define the concordance crosscap number of a knot as the minimum crosscap
number among all the knots concordant to the knot. The four-dimensional
crosscap number is the minimum first Betti number of non-orientable surfaces
smoothly embedded in 4-dimensional ball, bounding the knot. Clearly the
4-dimensional crosscap number is smaller than or equal to the concordance
crosscap number. We construct two infinite sequences of knots to explain the
gap between the two. In particular, the knot $7_4$ is one of the examples.
