Consider the universe of discourse
𝑈
=
𝑅
and binary relation

𝐿
=
{
(
𝑎
,
𝑏
)
|
𝑎
,
𝑏
∈
𝑅
,
𝑎

Question

Consider the universe of discourse 
𝑈
=
𝑅
 and binary relation

𝐿
=
{
(
𝑎
,
𝑏
)
|
𝑎
,
𝑏
∈
𝑅
,
𝑎
<
𝑏
}
.

Which one of the following wffs is true in this model?

BlackTom AI · Accepted Answer

The question provides a universe U = R and a binary relation L = { (a,b) | a,b in R, a < b }. We are asked which wff is true in this model, but the provided data includes an answer that is a single universal implication and there are no listed answer options to compare against.

First, restating the given wff in plain terms: For all x, for all y, for all z, if L(x,y) and L(y,z) then not L(z,x). Translating L into the usual less-than relation, this becomes: For all real numbers x, y, z, if x < y and y < z, then not (z < x).

Now evaluate the truth of this statement under the standard real numbers order:
- From x < y and y < z, by transitivity of < we have x < z.
- The conclusion not (z < x) means z is not less than x. Since we already know x < z, it certainly cannot be that z < x; thus not (z < x) holds.
- Therefore the implication (x < y ∧ y < z) ⇒ (not (z < x)) is true for this particular triple (x,y,z).

This reasoning holds for all real x, y, z, because the property x < y ∧ y < z always yields x < z, which in turn guarantees not (z < x). Consequently, the universal wff is satisfied in the model defined by U = R with L as the usual less-than relation.

Regarding the missing answer options: since none are provided, there is nothing to evaluate against the other choices. In a typical multiple-choice setup, you would compare this wff with each alternative to determine which is true in the model; here the sole wff described is indeed true, but without the option set, we cannot label a specific option as correct within the given format.

If an explicit set of options were present, the analysis would proceed by testing each option against the same reasoning: identify the logical form, substitute L as <, and assess whether the consequent necessarily follows from the antecedent for all real x, y, z. The key checks would involve preserving the transitivity of < and recognizing that x < y and y < z logically force x < z, which makes not (z < x) true.

Consider the universe of discourse 𝑈 = 𝑅 and binary relation 𝐿 = { ( 𝑎 , 𝑏 ) | 𝑎 , 𝑏 ∈ 𝑅 , 𝑎 < 𝑏 } . Which one of the following wffs is true in this model?Single choice

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