(d) : Given, 𝐴 = {1,2,3,4,5,6,7} \n\nand 𝑅 = {(𝑥, 𝑦) ∈ 𝐴 × 𝐴∶ 𝑥 + 𝑦 = 7} \n\nFor Reflexive : Let 𝑦 = 𝑥 \n\nSo, 𝑥 + 𝑥 = 7 ⇒ 𝑥 = 7, which is not possible. \n\n2 \n\nSo, given relation is not reflexive. \n\nFor symmetric : 𝑥𝑅𝑦 ⇒ 𝑥 + 𝑦 = 7 \n\n⇒ 𝑦 + 𝑥 = 7 ∴ 𝑥𝑅𝑦 ⇒ 𝑦𝑅𝑥 \n\nHence, given relation is symmetric. \n\nFor transitive : Let 𝑥𝑅𝑦 and 𝑦𝑅𝑧 ⇒ 𝑥 + 𝑦 = 7 and 𝑦 + 𝑧 = 7 \n\nBut it does not implies that, 𝑥 + 𝑧 = 7 \n\nSo, 𝑅 is not transitive. \n\n