MA 1971 Glossary

Question 1

The power set of a set A is…

Answer 1

the set of all subsets of A

Question 2

If a and b are integers, we say that a divides b if

Answer 2

there exists k ∈ Z such that b=ak

Question 3

A real number x is rational if…

Answer 3

there are integers a and b with b≠0 such that x=a/b.

Question 4

A relation R on a set X is reflexive if…

Answer 4

for all x∈ X, xRx

Question 5

A relation R on a set X is symmetric if…

Answer 5

for all x, y ∈ X, if xRy then yRx.

Question 6

A relation R on a set X is transitive if…

Answer 6

or all x, y, z ∈ X, if xRy and yRz, then xRz.

Question 7

A relation R on a set X is connected if…

Answer 7

for all x, y ∈ X, if x≠ y then xRy or yRx.

Question 8

If m ∈ N and a, b ∈ Z, we say that a is congruent to b modulo m if…

Answer 8

m divides a − b.

Question 9

Proof. Assume that X2 = d has no rational solutions, and, for the sake of con-
tradiction, that X2 = n2d does admit a rational solution. Say r ∈ Q satisfies r2 = n2d.
Writing r = a/b with a, b ∈ Z and b ̸ = 0, we have a2/b2 = n2d. Since n ̸ = 0, it follows that
a2/(b2n2) = d; equivalently, (a/(bn))2 = d. But this means that a/(bn) is a rational solution
of X2 = d, contrary to assumption. Therefore X2 = n2d has no rational solutions. □
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