1. **Problem Statement:** Given the set $A = \{2, 3, 4, 5\}$, define a relation $T$ on $A$ such that for every $x, y \in A$, $x T y$ if and only if $4 \mid (x + y)$, meaning $x + y$ is divisible by 4. 2. **Goal:** Find the Cartesian product $A \times A$ and determine which pairs $(x,y)$ satisfy the relation $T$. 3. **Step 1: Cartesian Product $A \times A$** The Cartesian product $A \times A$ is the set of all ordered pairs where the first and second elements are from $A$: $$ A \times A = \{(2,2), (2,3), (2,4), (2,5), (3,2), (3,3), (3,4), (3,5), (4,2), (4,3), (4,4), (4,5), (5,2), (5,3), (5,4), (5,5)\} $$ 4. **Step 2: Check divisibility by 4 for each pair** We check each pair $(x,y)$ to see if $4 \mid (x + y)$: - $(2,2): 2+2=4$, divisible by 4, so $(2,2) \in T$ - $(2,3): 2+3=5$, not divisible by 4 - $(2,4): 2+4=6$, not divisible by 4 - $(2,5): 2+5=7$, not divisible by 4 - $(3,2): 3+2=5$, no - $(3,3): 3+3=6$, no - $(3,4): 3+4=7$, no - $(3,5): 3+5=8$, yes, $(3,5) \in T$ - $(4,2): 4+2=6$, no - $(4,3): 4+3=7$, no - $(4,4): 4+4=8$, yes, $(4,4) \in T$ - $(4,5): 4+5=9$, no - $(5,2): 5+2=7$, no - $(5,3): 5+3=8$, yes, $(5,3) \in T$ - $(5,4): 5+4=9$, no - $(5,5): 5+5=10$, no 5. **Step 3: Final relation set $T$** $$ T = \{(2,2), (3,5), (4,4), (5,3)\} $$ **Answer:** The relation $T$ on $A$ consists of the pairs where the sum is divisible by 4, which are $(2,2)$, $(3,5)$, $(4,4)$, and $(5,3)$.