Here is a question.Please help me out.

Consider a two-person household with arbitrary preferences _ 1 and 12, which satisfy all the axioms we imposed on individual preferences. Note that, unlike the assignment problem, we do not specify any utility function for these individuals. Sometimes these two persons need to make joint decisions. They decide very simply: for any two alternatives, (x1, y1) and (x2, y2), they vote. If they both pick the same alternative, say (x1, y1), we say that household prefers (x1, y1). If one member of the household prefers (1, y1) and the other is indifferent, then we say the household prefers (x1, y1). If they pick different alternatives (say person 1 picks (x1, y1) and person 2 picks (x2, y2) ), then we say that household is indifferent between (x1, y1) and (X2, 2). Similarly, if they are both indifferent, we say that the household is indifferent. The resulting choice between two bundles defines the preferences of the household, > H. In lectures, we defined transitivity using "no-worse-than" (_) preference relation. If you prefer, you can use the following alternative definition of transitivity: preferences are transitive if, whenever (x1, yl) > (x2, y2) and (x2, y2) > (X3, 93), then (21, y1) > (23, 93). Prove that the preferences of the household are transitive or give a counter-example showing that preferences are not transitive