Solution
= Solution
The <real projective plane> has <fundamental group> $C_2=\langle a\mid a^2=1\rangle$, while the <circle> has <fundamental group> $\mathbb Z=\langle b\rangle$. Apply the <Seifert-van Kampen theorem> to open neighborhoods of the two wedge summands whose overlap is contractible. This gives
$$
\boxed{\pi_1(\mathbb {RP}^2\vee S^1,x)\cong C_2*\mathbb Z=\langle a,b\mid a^2=1\rangle.}
$$
Changing the basepoint within this path-connected space gives an isomorphic group.