91Ó°ÊÓ

Show a \(5 \times 5\) deficient board that is impossible to tile with trominoes. Explain why your board cannot be tiled with trominoes.

Prove that the product of two consecutive integers is even.

If the statement is true, prove it; otherwise, give a counterexample. The sets \(X, Y,\) and \(Z\) are subsets of a universal set \(U\). Assume that the universe for Cartesian products is \(U \times U\). \(X \cap(Y \times Z)=(X \cap Y) \times(X \cap Z)\) for all sets \(X, Y,\) and \(Z\).

A 3D-septomino is a three-dimensional \(2 \times 2 \times 2\) cube with one \(1 \times 1 \times 1\) corner cube removed. \(A\) deficient cube is \(a k \times k \times k\) cube with one \(1 \times 1 \times 1\) cube removed. Prove that if a \(k \times k \times k\) deficient cube can be tiled by 3D-septominoes, then 7 divides one of \(k-1, k-2, k-4\).

Prove that for all sets \(A\) and \(B, A \subseteq B\) if and only if \(\bar{B} \subseteq \bar{A}\).

The ordered pair \((a, b)\) can be defined in terms of sets as $$ (a, b)=\\{\\{a\\},\\{a, b\\}\\} $$ Taking the preceding equation as the definition of ordered pair, prove that \((a, b)=(c, d)\) if and only if \(a=c\) and \(b=d\).

Prove that the following are equivalent for sets \(A, B,\) and \(C\) : $$ \text { (a) } A \cup B=U $$ (b) \(\bar{A} \cap \bar{B}=\varnothing\) (c) \(\bar{A} \subset B\), where \(U\) is a universal set.

Suppose that \(n>1\) people are positioned in a field (Euclidean plane) so that each has a unique nearest neighbor: Suppose further that each person has a pie that is hurled at the nearest neighbor: A survivor is a person that is not hit by a pie. Prove or disprove: If \(n\) is odd, one of two persons farthest apart is a survivor.

\(\Delta\) denotes the symmetric difference operator defined as \(A \triangle B=(A \cup B)-(A \cap B),\) where \(A\) and \(B\) are sets. Is \(\Delta\) associative? If so, prove it; otherwise, give a counterexample.