Examples of places where I see this used:
Let * be a binary operation in S and let H be a subset of S… the binary operation on H given by restricting * to H is the induced operation of * on H
For a subgroup, it is not sufficient that "the set of one group be a subset of another, but also that the group operation on the subset be the induced operation that assigns the same element to each ordered pair from this subset as is assigned by the group operation on the whole set"
I just really don't understand what an induced operation is supposed to be.
Is this an accurate understanding?: If you have the binary structure $\langle S,* \rangle$, then the induced operation is if you make another binary structure with set S' and let the operation be the same * as in the first binary structure. In other words, is it accurate to say that the operation * in the binary structure $\langle S',* \rangle$ is the induced operation of $\langle S,* \rangle$, but the operation *' in $\langle S',*' \rangle$ is not?
All quotes from the book "A First Course in Abstract Algebra" by John Fraleigh, 7th edition
Best Answer
Your suggestion is correct, except that you forgot to say what $S'$ has to do with $S$. And it is this: $S'$ is a subset of $S$. For instance, sum on $\mathbb R$ does not induce an operation in $(-1,1)$ since, in general the sum of two elements of $(-1,1)$ is not an element of $(-1,1)$.
There is another case of induced operations, which appears in the context of quotients of algebraic structures, but I don't know whether or not you are familar with this.