Theory question 101
Binary OperationsA binary operation * is defined on the set T = {−2, −1, 1, 2} by p * q = p2 + 2pq − q2, where p, q ∈ T.
(a) Copy and complete the table (row p, column q; blanks to be filled):
Row −2: q = −2: __, q = −1: 7, q = 1: __, q = 2: −8
Row −1: q = −2: __, q = −1: 2, q = 1: −2, q = 2: __
Row 1: q = −2: −7, q = −1: __, q = 1: __, q = 2: 1
Row 2: q = −2: __, q = −1: −1, q = 1: __, q = 2: __
(b) Using the table in 1(a), find the value of p such that (−2 * p) * 2 = −7.
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(a) Complete table (columns q = −2, −1, 1, 2):
Row −2: 8, 7, −1, −8
Row −1: 1, 2, −2, −7
Row 1: −7, −2, 2, 1
Row 2: −8, −1, 7, 8
(e.g. −2 * −2 = 4 + 8 − 4 = 8; 1 * −1 = 1 − 2 − 1 = −2.)
(b) From the column q = 2, a * 2 = −7 only when a = −1. So −2 * p = −1, and row −2 gives −1 at q = 1. Hence p = 1.
Substitute each pair carefully, keeping track of the minus signs.