Which statements are true about inventory systems?$$

$1.$ For payment to a supplier within the discount period, Purchase discounts is credited under the periodic system and Inventory is credited under the perpetual system.

$2.$ To record shipping cost, both periodic and perpetual systems debit Freight$-$in$.$

$3.$ To record merchandise purchases, Purchases is debited under the periodic system and Inventory is debited under the perpetual system.

$4.$ Both periodic and perpetual inventory systems involve units of inventory.

$5.$ To record merchandise returned to a supplier, Purchases is credited under the periodic system and Inventory is credited under the perpetual system.