The set $D_1$ consists of three students: Garth, who is 5 feet 11 inches tall; Erin, who is 5 feet 6 inches tall; and Marty, who is 6 feet tall. The set $D_2$ consists of four students: Dale, who is 5 feet 11 inches tall; Garth, who is 5 feet 11 inches tall; Erin, who is 5 feet 6 inches tall; and Mary, who is 6 feet tall. The set $D_3$ consists of one student: Dale, who is 6 feet tall. The set $D_4$ consists of three students: Part, Sandy, and Gal, each of whom is 5 feet 11 inches tall. $T_1(x,y)$ is the propositional function "x is taller than y." Tell whether each proposition is true or false if the domain of discourse is $D_i\times D_j$ for the given values of i and j.

- $\forall x\forall yT_1(x,y)$
- $\forall x\exists yT_1(x,y)$
- $\exists x\forall yT_1(x,y)$
- $\exists x\exists yT_1(x,y)$

i=4, j=1

Solution

VerifiedGiven:

$T_1(x,y)$ =$x$ is taller than $y$

$D_1$=$\{$Garth (5 ft 11 in), Erin (5 ft 6 in), Marty (6 ft)$\}$

$D_2$=$\{$Dale (6 ft); Garth (5 ft 11 in), Erin (5 ft 6 in), Marty (6 ft)$\}$

$D_3$=$\{$Dale (6 ft)$\}$

$D_4$=$\{$Pat (5 ft 11 in), Sandy (5 ft 11 in), Gale (5 ft 11 in)$\}$

$i=4$

$j=1$

$\textbf{Cartesian product}$ of $A$ and $B$: $A\times B=\{(a,b)|a\in A\wedge b\in B\}$

Let us determine all elements of $D_i\times D_j$:

$\begin{align*} D_4\times D_1=\{&(Pat, Garth), (Pat, Erin), (Pat, Marty), \\ &(Sandy, Garth), (Sandy, Erin), (Sandy, Marty), \\ &(Gale, Garth), (Gale, Erin), (Gale, Marty)\} \end{align*}$

$\forall x\forall y T_1(x,y)$: Everyone is taller than everyone. This statement is $\textbf{false}$, because we note that Pat is not taller than Marty.

$\forall x\exists y T_1(x,y)$: Everyone is taller than somebody. This statement is $\textbf{true}$, because we note that Pat, Sandy and Gale are all taller than Erin.

$\exists x\forall y T_1(x,y)$: Somebody is taller than everyone. This statement is $\textbf{false}$, because we note that Pat, Sandy and Gale are all not taller than Marty.

$\exists x\exists y T_1(x,y)$: Someone is taller than someone. This statement is $\textbf{true}$, because we note that Pat is taller than Erin.

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