(a) If I'm reading the table correctly, by definition of *, we have
<em>a</em> * <em>a</em> = <em>b</em>
<em>a</em> * <em>b</em> = <em>b</em>
<em>b</em> * <em>a</em> = a
<em>b</em> * <em>b</em> = <em>a</em>
<em />
(b) Yes, * is a binary operation. A binary operation is a function * that maps elements from <em>A</em> × <em>A</em> to <em>A</em>, where
<em>A</em> × <em>A</em> = {{<em>a</em>, <em>b</em>} | <em>a</em> ∈ <em>A</em> and <em>b</em> ∈ <em>A</em>}
In this case,
<em>A</em> × <em>A</em> = {{<em>a</em>, <em>a</em>}, {<em>a</em>, <em>b</em>}, {<em>b</em>, <em>a</em>}, {<em>b</em>, <em>b</em>}}
and * is defined such that each of these pairs gets mapped to either <em>a</em> or <em>b</em>, both elements of <em>A</em>. In other words, <em>A</em> is <u>closed under *</u>.
(c) The domain is <em>A</em> × <em>A</em> and the co-domain is <em>A</em>.